Database - belyi_galmaps (label)

Formats: - HTML - YAML - JSON - 2024-09-21T00:50:21.269453 - next page

Query: /api/belyi_galmaps/?_offset=0

Column	Type
BelyiDB_label	text
BelyiDB_plabel	text
a_s	smallint
abc	smallint[]
aut_group	jsonb
b_s	smallint
base_field	numeric[]
base_field_label	text
c_s	smallint
curve	text
curve_label	text
deg	smallint
embeddings	jsonb
friends	text[]
g	smallint
geomtype	text
group	text
group_num	smallint
is_primitive	boolean
label	text
lambdas	jsonb
map	text
models	jsonb
moduli_field	numeric[]
moduli_field_label	text
orbit_size	smallint
pass_size	smallint
plabel	text
plane_constant	text
plane_map_constant_factored	text
plane_model	text
plane_model_latex	text
primitivization	text
specializations	jsonb
triples	jsonb
triples_cyc	jsonb

0. label: 1T1-1_1_1-a
   {'BelyiDB_label': '1T1-[1,1,1]-1-1-1-g0-a', 'BelyiDB_plabel': '1T1-[1,1,1]-1-1-1-g0', 'a_s': 1, 'abc': [1, 1, 1], 'aut_group': [[1]], 'b_s': 1, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 1, 'curve': 'PP1', 'deg': 1, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '1T1', 'group_num': 1, 'is_primitive': True, 'label': '1T1-1_1_1-a', 'lambdas': [[1], [1], [1]], 'map': 'x', 'orbit_size': 1, 'pass_size': 1, 'plabel': '1T1-1_1_1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't + x', 'plane_model_latex': 't + x', 'primitivization': '1T1-1_1_1-a', 'triples': [[[1], [1], [1]]], 'triples_cyc': [['()', '()', '()']]}
1. label: 2T1-2_2_1.1-a
   {'BelyiDB_label': '2T1-[2,2,1]-2-2-11-g0-a', 'BelyiDB_plabel': '2T1-[2,2,1]-2-2-11-g0', 'a_s': 1, 'abc': [2, 2, 1], 'aut_group': [[2, 1]], 'b_s': 2, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 2, 'curve': 'PP1', 'deg': 2, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '2T1', 'group_num': 1, 'is_primitive': True, 'label': '2T1-2_2_1.1-a', 'lambdas': [[2], [2], [1, 1]], 'map': '-1/(x^2-1)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '2T1-2_2_1.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't + x^2', 'plane_model_latex': 't + x^{2}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[2, 1], [2, 1], [1, 2]]], 'triples_cyc': [['(1,2)', '(1,2)', '()']]}
2. label: 3T1-3_3_1.1.1-a
   {'BelyiDB_label': '3T1-[3,3,1]-3-3-111-g0-a', 'BelyiDB_plabel': '3T1-[3,3,1]-3-3-111-g0', 'a_s': 1, 'abc': [3, 3, 1], 'aut_group': [[2, 3, 1]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 3, 'curve': 'PP1', 'deg': 3, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '3T1', 'group_num': 1, 'is_primitive': True, 'label': '3T1-3_3_1.1.1-a', 'lambdas': [[3], [3], [1, 1, 1]], 'map': '-1/(x^3-1)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '3T1-3_3_1.1.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't + x^3', 'plane_model_latex': 't + x^{3}', 'primitivization': '3T1-3_3_1.1.1-a', 'triples': [[[2, 3, 1], [3, 1, 2], [1, 2, 3]]], 'triples_cyc': [['(1,2,3)', '(1,3,2)', '()']]}
3. label: 3T1-3_3_3-a
   {'BelyiDB_label': '3T1-[3,3,3]-3-3-3-g1-a', 'BelyiDB_plabel': '3T1-[3,3,3]-3-3-3-g1', 'a_s': 3, 'abc': [3, 3, 3], 'aut_group': [[2, 3, 1]], 'b_s': 3, 'base_field': [1, -1, 1], 'base_field_label': '2.0.3.1', 'c_s': 3, 'curve': 'y^2=x^3+1', 'curve_label': '2.0.3.1-144.1-CMa1', 'deg': 3, 'embeddings': [[0.5, 0.8660254037844387]], 'friends': ['EllipticCurve/2.0.3.1/144.1/CMa/1'], 'g': 1, 'geomtype': 'E', 'group': '3T1', 'group_num': 1, 'is_primitive': True, 'label': '3T1-3_3_3-a', 'lambdas': [[3], [3], [3]], 'map': '-1/2*y+1/2', 'orbit_size': 1, 'pass_size': 1, 'plabel': '3T1-3_3_3', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't^2 - t + 2*x^3', 'plane_model_latex': 't^{2} - t + 2 x^{3}', 'primitivization': '3T1-3_3_3-a', 'triples': [[[2, 3, 1], [2, 3, 1], [2, 3, 1]]], 'triples_cyc': [['(1,2,3)', '(1,2,3)', '(1,2,3)']]}
4. label: 3T2-3_2.1_2.1-a
   {'BelyiDB_label': '3T2-[3,2,2]-3-21-21-g0-a', 'BelyiDB_plabel': '3T2-[3,2,2]-3-21-21-g0', 'a_s': 2, 'abc': [3, 2, 2], 'aut_group': [[1, 2, 3]], 'b_s': 2, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 3, 'curve': 'PP1', 'deg': 3, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '3T2', 'group_num': 2, 'is_primitive': True, 'label': '3T2-3_2.1_2.1-a', 'lambdas': [[3], [2, 1], [2, 1]], 'map': '1/2/(x^3-3/4*x+1/4)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '3T2-3_2.1_2.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 'x^3*t + (-3*x - 2)', 'plane_model_latex': 'x^{3} t + \\left(-3 x - 2\\right)', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[2, 3, 1], [1, 3, 2], [2, 1, 3]]], 'triples_cyc': [['(1,2,3)', '(2,3)', '(1,2)']]}
5. label: 4T1-4_4_1.1.1.1-a
   {'BelyiDB_label': '4T1-[4,4,1]-4-4-1111-g0-a', 'BelyiDB_plabel': '4T1-[4,4,1]-4-4-1111-g0', 'a_s': 1, 'abc': [4, 4, 1], 'aut_group': [[2, 3, 4, 1]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 4, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '4T1', 'group_num': 1, 'is_primitive': False, 'label': '4T1-4_4_1.1.1.1-a', 'lambdas': [[4], [4], [1, 1, 1, 1]], 'map': '-1/(x^4-1)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '4T1-4_4_1.1.1.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't + x^4', 'plane_model_latex': 't + x^{4}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[2, 3, 4, 1], [4, 1, 2, 3], [1, 2, 3, 4]]], 'triples_cyc': [['(1,2,3,4)', '(1,4,3,2)', '()']]}
6. label: 4T1-4_4_2.2-a
   {'BelyiDB_label': '4T1-[4,4,2]-4-4-22-g1-a', 'BelyiDB_plabel': '4T1-[4,4,2]-4-4-22-g1', 'a_s': 2, 'abc': [4, 4, 2], 'aut_group': [[2, 3, 4, 1]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'y^2=x^3-x', 'curve_label': '32.a3', 'deg': 4, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/32/a/3'], 'g': 1, 'geomtype': 'E', 'group': '4T1', 'group_num': 1, 'is_primitive': False, 'label': '4T1-4_4_2.2-a', 'lambdas': [[4], [4], [2, 2]], 'map': 'x^2/(x^2-1)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '4T1-4_4_2.2', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '(-1)*t^2 + t + x^4', 'plane_model_latex': '\\left(-1\\right) t^{2} + t + x^{4}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[2, 3, 4, 1], [2, 3, 4, 1], [3, 4, 1, 2]]], 'triples_cyc': [['(1,2,3,4)', '(1,2,3,4)', '(1,3)(2,4)']]}
7. label: 4T2-2.2_2.2_2.2-a
   {'BelyiDB_label': '4T2-[2,2,2]-22-22-22-g0-a', 'BelyiDB_plabel': '4T2-[2,2,2]-22-22-22-g0', 'a_s': 2, 'abc': [2, 2, 2], 'aut_group': [[3, 4, 1, 2], [4, 3, 2, 1]], 'b_s': 2, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 2, 'curve': 'PP1', 'deg': 4, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '4T2', 'group_num': 2, 'is_primitive': False, 'label': '4T2-2.2_2.2_2.2-a', 'lambdas': [[2, 2], [2, 2], [2, 2]], 'map': '(-x^4+2*x^3-x^2)/(x^2-x+1/4)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '4T2-2.2_2.2_2.2', 'plane_constant': '4', 'plane_map_constant_factored': '\\frac{1}{2^{2}} ', 'plane_model': 'x^2*t + (x - 1)^2*(x + 1)^2', 'plane_model_latex': 'x^{2} t + \\left(x - 1\\right)^{2} \\left(x + 1\\right)^{2}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[4, 3, 2, 1], [2, 1, 4, 3], [3, 4, 1, 2]]], 'triples_cyc': [['(1,4)(2,3)', '(1,2)(3,4)', '(1,3)(2,4)']]}
8. label: 4T3-4_2.2_2.1.1-a
   {'BelyiDB_label': '4T3-[4,2,2]-4-22-211-g0-a', 'BelyiDB_plabel': '4T3-[4,2,2]-4-22-211-g0', 'a_s': 2, 'abc': [4, 2, 2], 'aut_group': [[3, 4, 1, 2]], 'b_s': 2, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 4, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '4T3', 'group_num': 3, 'is_primitive': False, 'label': '4T3-4_2.2_2.1.1-a', 'lambdas': [[4], [2, 2], [2, 1, 1]], 'map': '-1/4/(x^4-x^2)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '4T3-4_2.2_2.1.1', 'plane_constant': '1/4', 'plane_map_constant_factored': '2^{2}', 'plane_model': 't + x^2*(x - 1)*(x + 1)', 'plane_model_latex': 't + x^{2} \\left(x - 1\\right) \\left(x + 1\\right)', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[2, 3, 4, 1], [2, 1, 4, 3], [3, 2, 1, 4]]], 'triples_cyc': [['(1,2,3,4)', '(1,2)(3,4)', '(1,3)']]}
9. label: 4T4-3.1_3.1_2.2-a
   {'BelyiDB_label': '4T4-[3,3,2]-31-31-22-g0-a', 'BelyiDB_plabel': '4T4-[3,3,2]-31-31-22-g0', 'a_s': 2, 'abc': [3, 3, 2], 'aut_group': [[1, 2, 3, 4]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 3, 'curve': 'PP1', 'deg': 4, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '4T4', 'group_num': 4, 'is_primitive': True, 'label': '4T4-3.1_3.1_2.2-a', 'lambdas': [[3, 1], [3, 1], [2, 2]], 'map': 'x/(x^4-3*x^3+15/8*x^2+9/16*x+9/256)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '4T4-3.1_3.1_2.2', 'plane_constant': '1/64', 'plane_map_constant_factored': '2^{6}', 'plane_model': '(8*x - 1)*t + x^3*(x + 1)', 'plane_model_latex': '\\left(8 x - 1\\right) t + x^{3} \\left(x + 1\\right)', 'primitivization': '4T4-3.1_3.1_2.2-a', 'triples': [[[2, 3, 1, 4], [2, 4, 3, 1], [3, 4, 1, 2]]], 'triples_cyc': [['(1,2,3)', '(1,2,4)', '(1,3)(2,4)']]}
10. label: 4T4-3.1_3.1_3.1-a
    {'BelyiDB_label': '4T4-[3,3,3]-31-31-31-g0-a', 'BelyiDB_plabel': '4T4-[3,3,3]-31-31-31-g0', 'a_s': 3, 'abc': [3, 3, 3], 'aut_group': [[1, 2, 3, 4]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 3, 'curve': 'PP1', 'deg': 4, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'E', 'group': '4T4', 'group_num': 4, 'is_primitive': True, 'label': '4T4-3.1_3.1_3.1-a', 'lambdas': [[3, 1], [3, 1], [3, 1]], 'map': '(-4*x^4+4*x^3)/(x-1/4)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '4T4-3.1_3.1_3.1', 'plane_constant': '1/16', 'plane_map_constant_factored': '2^{4}', 'plane_model': '(4*x - 1)*t + x^3*(x - 1)', 'plane_model_latex': '\\left(4 x - 1\\right) t + x^{3} \\left(x - 1\\right)', 'primitivization': '4T4-3.1_3.1_3.1-a', 'triples': [[[2, 3, 1, 4], [3, 2, 4, 1], [1, 4, 2, 3]]], 'triples_cyc': [['(1,2,3)', '(1,3,4)', '(2,4,3)']]}
11. label: 4T5-4_3.1_2.1.1-a
    {'BelyiDB_label': '4T5-[4,3,2]-4-31-211-g0-a', 'BelyiDB_plabel': '4T5-[4,3,2]-4-31-211-g0', 'a_s': 2, 'abc': [4, 3, 2], 'aut_group': [[1, 2, 3, 4]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 4, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '4T5', 'group_num': 5, 'is_primitive': True, 'label': '4T5-4_3.1_2.1.1-a', 'lambdas': [[4], [3, 1], [2, 1, 1]], 'map': '1/243/(x^4-26/9*x^3+55/18*x^2-25/18*x+11/48)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '4T5-4_3.1_2.1.1', 'plane_constant': '-256/27', 'plane_map_constant_factored': '\\frac{3^{3}}{2^{8}} ', 'plane_model': '(-1)*(x - 1)*t + x^4', 'plane_model_latex': '\\left(-1\\right) \\left(x - 1\\right) t + x^{4}', 'primitivization': '4T5-4_3.1_2.1.1-a', 'triples': [[[2, 3, 4, 1], [1, 4, 2, 3], [2, 1, 3, 4]]], 'triples_cyc': [['(1,2,3,4)', '(2,4,3)', '(1,2)']]}
12. label: 4T5-4_4_3.1-a
    {'BelyiDB_label': '4T5-[4,4,3]-4-4-31-g1-a', 'BelyiDB_plabel': '4T5-[4,4,3]-4-4-31-g1', 'a_s': 3, 'abc': [4, 4, 3], 'aut_group': [[1, 2, 3, 4]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'y^2=x^3+47/768*x+2359/55296', 'curve_label': '48.a6', 'deg': 4, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/48/a/6'], 'g': 1, 'geomtype': 'H', 'group': '4T5', 'group_num': 5, 'is_primitive': True, 'label': '4T5-4_4_3.1-a', 'lambdas': [[4], [4], [3, 1]], 'map': '27/64/(x^4-13/12*x^3-155/384*x^2-1225/27648*x-8375/5308416)*y+(-27/64*x^2+9/512*x+1401/16384)/(x^4-13/12*x^3-155/384*x^2-1225/27648*x-8375/5308416)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '4T5-4_4_3.1', 'plane_constant': '27/4', 'plane_map_constant_factored': '\\frac{2^{2}}{3^{3}} ', 'plane_model': 't^2 + (2*x^2 - 4*x + 9)*t + x^4', 'plane_model_latex': 't^{2} + \\left(2 x^{2} - 4 x + 9\\right) t + x^{4}', 'primitivization': '4T5-4_4_3.1-a', 'triples': [[[2, 3, 4, 1], [3, 1, 4, 2], [3, 2, 4, 1]]], 'triples_cyc': [['(1,2,3,4)', '(1,3,4,2)', '(1,3,4)']]}
13. label: 5T1-5_5_1.1.1.1.1-a
    {'BelyiDB_label': '5T1-[5,5,1]-5-5-11111-g0-a', 'BelyiDB_plabel': '5T1-[5,5,1]-5-5-11111-g0', 'a_s': 1, 'abc': [5, 5, 1], 'aut_group': [[2, 3, 4, 5, 1]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '5T1', 'group_num': 1, 'is_primitive': True, 'label': '5T1-5_5_1.1.1.1.1-a', 'lambdas': [[5], [5], [1, 1, 1, 1, 1]], 'map': '-1/(x^5-1)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T1-5_5_1.1.1.1.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't + x^5', 'plane_model_latex': 't + x^{5}', 'primitivization': '5T1-5_5_1.1.1.1.1-a', 'triples': [[[2, 3, 4, 5, 1], [5, 1, 2, 3, 4], [1, 2, 3, 4, 5]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,5,4,3,2)', '()']]}
14. label: 5T1-5_5_5-a
    {'BelyiDB_label': '5T1-[5,5,5]-5-5-5-g2-a', 'BelyiDB_plabel': '5T1-[5,5,5]-5-5-5-g2', 'a_s': 5, 'abc': [5, 5, 5], 'aut_group': [[2, 3, 4, 5, 1]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^6+2*x', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 2, 'geomtype': 'H', 'group': '5T1', 'group_num': 1, 'is_primitive': True, 'label': '5T1-5_5_5-a', 'lambdas': [[5], [5], [5]], 'map': 'x^2*y+x^5+1', 'orbit_size': 1, 'pass_size': 3, 'plabel': '5T1-5_5_5', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '(-2)*t^2 - 2*t + x^5', 'plane_model_latex': '\\left(-2\\right) t^{2} - 2 t + x^{5}', 'primitivization': '5T1-5_5_5-a', 'triples': [[[2, 3, 4, 5, 1], [4, 5, 1, 2, 3], [2, 3, 4, 5, 1]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,4,2,5,3)', '(1,2,3,4,5)']]}
15. label: 5T1-5_5_5-b
    {'BelyiDB_label': '5T1-[5,5,5]-5-5-5-g2-b', 'BelyiDB_plabel': '5T1-[5,5,5]-5-5-5-g2', 'a_s': 5, 'abc': [5, 5, 5], 'aut_group': [[2, 3, 4, 5, 1]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^6-2*x', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 2, 'geomtype': 'H', 'group': '5T1', 'group_num': 1, 'is_primitive': True, 'label': '5T1-5_5_5-b', 'lambdas': [[5], [5], [5]], 'map': '1/2/x^3*y+1/2', 'orbit_size': 1, 'pass_size': 3, 'plabel': '5T1-5_5_5', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '2*t^2 - 2*t + x^5', 'plane_model_latex': '2 t^{2} - 2 t + x^{5}', 'primitivization': '5T1-5_5_5-b', 'triples': [[[2, 3, 4, 5, 1], [2, 3, 4, 5, 1], [4, 5, 1, 2, 3]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,2,3,4,5)', '(1,4,2,5,3)']]}
16. label: 5T1-5_5_5-c
    {'BelyiDB_label': '5T1-[5,5,5]-5-5-5-g2-c', 'BelyiDB_plabel': '5T1-[5,5,5]-5-5-5-g2', 'a_s': 5, 'abc': [5, 5, 5], 'aut_group': [[2, 3, 4, 5, 1]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=4*x^5+1/4', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 2, 'geomtype': 'H', 'group': '5T1', 'group_num': 1, 'is_primitive': True, 'label': '5T1-5_5_5-c', 'lambdas': [[5], [5], [5]], 'map': '-1/4/x^5*y-1/8/x^5', 'orbit_size': 1, 'pass_size': 3, 'plabel': '5T1-5_5_5', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '(-1)*t^2 + t + 4*x^5', 'plane_model_latex': '\\left(-1\\right) t^{2} + t + 4 x^{5}', 'primitivization': '5T1-5_5_5-c', 'triples': [[[2, 3, 4, 5, 1], [3, 4, 5, 1, 2], [3, 4, 5, 1, 2]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,3,5,2,4)', '(1,3,5,2,4)']]}
17. label: 5T2-5_2.2.1_2.2.1-a
    {'BelyiDB_label': '5T2-[5,2,2]-5-221-221-g0-a', 'BelyiDB_plabel': '5T2-[5,2,2]-5-221-221-g0', 'a_s': 2, 'abc': [5, 2, 2], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 2, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '5T2', 'group_num': 2, 'is_primitive': True, 'label': '5T2-5_2.2.1_2.2.1-a', 'lambdas': [[5], [2, 2, 1], [2, 2, 1]], 'map': '1/8/(x^5-5/4*x^3+5/16*x+1/16)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T2-5_2.2.1_2.2.1', 'plane_constant': '-4/125', 'plane_map_constant_factored': '\\frac{5^{3}}{2^{2}} ', 'plane_model': 'x^5*t + (x^2 - 5*x + 5)^2', 'plane_model_latex': 'x^{5} t + \\left(x^{2} - 5 x + 5\\right)^{2}', 'primitivization': '5T2-5_2.2.1_2.2.1-a', 'triples': [[[2, 3, 4, 5, 1], [3, 2, 1, 5, 4], [4, 3, 2, 1, 5]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,3)(4,5)', '(1,4)(2,3)']]}
18. label: 5T3-4.1_4.1_2.2.1-a
    {'BelyiDB_label': '5T3-[4,4,2]-41-41-221-g0-a', 'BelyiDB_plabel': '5T3-[4,4,2]-41-41-221-g0', 'a_s': 2, 'abc': [4, 4, 2], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 4, 'base_field': [1, 0, 1], 'base_field_label': '2.0.4.1', 'c_s': 4, 'curve': 'PP1', 'deg': 5, 'embeddings': [[0.0, -1.0], [0.0, 1.0]], 'friends': [], 'g': 0, 'geomtype': 'E', 'group': '5T3', 'group_num': 3, 'is_primitive': True, 'label': '5T3-4.1_4.1_2.2.1-a', 'lambdas': [[4, 1], [4, 1], [2, 2, 1]], 'map': '(1/4107*(136*nu+4623)*x^5+1/37*(136*nu+4623)*x^4)/(x^5+(-40*nu+55)*x^4+(-13236*nu-12048)*x^3+(-1006992*nu-709956)*x^2+(67346586*nu-36186777)*x+(7475471046*nu-4016732247))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '5T3-4.1_4.1_2.2.1', 'plane_constant': '1/3125*(41*nu - 38)', 'plane_map_constant_factored': '1 (-41 \\nu - 38)', 'plane_model': '(-1)*((-4*nu + 3)*x - 1)*t + nu*x^4*(x + 1)', 'plane_model_latex': '\\left(\\left(4 \\nu - 3\\right) x + 1\\right) t + \\nu x^{4} \\left(x + 1\\right)', 'primitivization': '5T3-4.1_4.1_2.2.1-a', 'triples': [[[4, 1, 3, 5, 2], [1, 3, 5, 2, 4], [4, 3, 2, 1, 5]], [[2, 5, 3, 1, 4], [5, 3, 1, 4, 2], [4, 3, 2, 1, 5]]], 'triples_cyc': [['(1,4,5,2)', '(2,3,5,4)', '(1,4)(2,3)'], ['(1,2,5,4)', '(1,5,2,3)', '(1,4)(2,3)']]}
19. label: 5T3-5_4.1_4.1-a
    {'BelyiDB_label': '5T3-[5,4,4]-5-41-41-g1-a', 'BelyiDB_plabel': '5T3-[5,4,4]-5-41-41-g1', 'a_s': 4, 'abc': [5, 4, 4], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 4, 'base_field': [1, 0, 1], 'base_field_label': '2.0.4.1', 'c_s': 5, 'curve': 'y^2=x^3-1/75*x-118/16875', 'curve_label': '2.0.4.1-1250.3-a3', 'deg': 5, 'embeddings': [[0.0, 1.0], [0.0, -1.0]], 'friends': ['EllipticCurve/2.0.4.1/1250.3/a/3'], 'g': 1, 'geomtype': 'H', 'group': '5T3', 'group_num': 3, 'is_primitive': True, 'label': '5T3-5_4.1_4.1-a', 'lambdas': [[5], [4, 1], [4, 1]], 'map': '(-256/3125*nu*x-1792/46875*nu)/(x^5-1/15*x^4+58/225*x^3+1/84375*(6912*nu-866)*x^2+1/6328125*(-13824*nu-61343)*x+1/2373046875*(-960768*nu+309599))*y+(256/3125*nu*x^2-512/234375*nu*x+1/87890625*(-35584*nu+294912))/(x^5-1/15*x^4+58/225*x^3+1/84375*(6912*nu-866)*x^2+1/6328125*(-13824*nu-61343)*x+1/2373046875*(-960768*nu+309599))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '5T3-5_4.1_4.1', 'plane_constant': '1/125*(-16*nu - 88)', 'plane_map_constant_factored': '\\frac{1}{2^{3}} (2 \\nu - 11)', 'plane_model': '(11*nu + 2)*x^5*t^2 + (8*nu*x^5 + 5*x - 1)*t + ((-nu + 2)*x - 2*nu)', 'plane_model_latex': '\\left(11 \\nu + 2\\right) x^{5} t^{2} + \\left(8 \\nu x^{5} + 5 x - 1\\right) t + \\nu \\left(\\left(-2 \\nu - 1\\right) x - 2\\right)', 'primitivization': '5T3-5_4.1_4.1-a', 'triples': [[[2, 3, 4, 5, 1], [1, 3, 5, 2, 4], [3, 1, 4, 2, 5]], [[2, 3, 4, 5, 1], [2, 5, 3, 1, 4], [2, 4, 1, 3, 5]]], 'triples_cyc': [['(1,2,3,4,5)', '(2,3,5,4)', '(1,3,4,2)'], ['(1,2,3,4,5)', '(1,2,5,4)', '(1,2,4,3)']]}
20. label: 5T4-5_2.2.1_3.1.1-a
    {'BelyiDB_label': '5T4-[5,2,3]-5-221-311-g0-a', 'BelyiDB_plabel': '5T4-[5,2,3]-5-221-311-g0', 'a_s': 2, 'abc': [5, 2, 3], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '5T4', 'group_num': 4, 'is_primitive': True, 'label': '5T4-5_2.2.1_3.1.1-a', 'lambdas': [[5], [2, 2, 1], [3, 1, 1]], 'map': '-2/9/(x^5-10/3*x^4+95/18*x^3-25/6*x^2+25/16*x-2/9)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T4-5_2.2.1_3.1.1', 'plane_constant': '1728', 'plane_map_constant_factored': '\\frac{1}{2^{6} \\cdot 3^{3}} ', 'plane_model': 'x^5*t - (-40*x^2 + 5*x - 1)', 'plane_model_latex': 'x^{5} t - \\left(-40 x^{2} + 5 x - 1\\right)', 'primitivization': '5T4-5_2.2.1_3.1.1-a', 'triples': [[[3, 5, 4, 2, 1], [1, 5, 4, 3, 2], [2, 3, 1, 4, 5]]], 'triples_cyc': [['(1,3,4,2,5)', '(2,5)(3,4)', '(1,2,3)']]}
21. label: 5T4-5_3.1.1_3.1.1-a
    {'BelyiDB_label': '5T4-[5,3,3]-5-311-311-g0-a', 'BelyiDB_plabel': '5T4-[5,3,3]-5-311-311-g0', 'a_s': 3, 'abc': [5, 3, 3], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '5T4', 'group_num': 4, 'is_primitive': True, 'label': '5T4-5_3.1.1_3.1.1-a', 'lambdas': [[5], [3, 1, 1], [3, 1, 1]], 'map': '2*x^5/(x^5-45/8*x^4+135/4*x^2-729/8)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T4-5_3.1.1_3.1.1', 'plane_constant': '1296', 'plane_map_constant_factored': '\\frac{1}{2^{4} \\cdot 3^{4}} ', 'plane_model': 'x^5*t + (-60*x^2 + 15*x - 1)', 'plane_model_latex': 'x^{5} t + \\left(-60 x^{2} + 15 x - 1\\right)', 'primitivization': '5T4-5_3.1.1_3.1.1-a', 'triples': [[[3, 5, 2, 1, 4], [2, 3, 1, 4, 5], [4, 2, 3, 5, 1]]], 'triples_cyc': [['(1,3,2,5,4)', '(1,2,3)', '(1,4,5)']]}
22. label: 5T4-5_5_2.2.1-a
    {'BelyiDB_label': '5T4-[5,5,2]-5-5-221-g1-a', 'BelyiDB_plabel': '5T4-[5,5,2]-5-5-221-g1', 'a_s': 2, 'abc': [5, 5, 2], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^3+211/375*x-6214/84375', 'curve_label': '50.b4', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/50/b/4'], 'g': 1, 'geomtype': 'H', 'group': '5T4', 'group_num': 4, 'is_primitive': True, 'label': '5T4-5_5_2.2.1-a', 'lambdas': [[5], [5], [2, 2, 1]], 'map': '(2048/3125*x-2048/46875)/(x^5-17/15*x^4+578/1125*x^3-120418/84375*x^2-137663/6328125*x-969431633/2373046875)*y+(-2048/3125*x^2-4096/234375*x-36800512/87890625)/(x^5-17/15*x^4+578/1125*x^3-120418/84375*x^2-137663/6328125*x-969431633/2373046875)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T4-5_5_2.2.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '4*x^5*t^2 + (-5*x - 2)*t - (-5*x - 2)', 'plane_model_latex': '4 x^{5} t^{2} + \\left(-5 x - 2\\right) t - \\left(-5 x - 2\\right)', 'primitivization': '5T4-5_5_2.2.1-a', 'triples': [[[2, 3, 4, 5, 1], [3, 1, 4, 5, 2], [4, 2, 5, 1, 3]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,3,4,5,2)', '(1,4)(3,5)']]}
23. label: 5T4-5_5_3.1.1-a
    {'BelyiDB_label': '5T4-[5,5,3]-5-5-311-g1-a', 'BelyiDB_plabel': '5T4-[5,5,3]-5-5-311-g1', 'a_s': 3, 'abc': [5, 5, 3], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^3-269/375*x+18962/16875', 'curve_label': '150.c3', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/150/c/3'], 'g': 1, 'geomtype': 'H', 'group': '5T4', 'group_num': 4, 'is_primitive': True, 'label': '5T4-5_5_3.1.1-a', 'lambdas': [[5], [5], [3, 1, 1]], 'map': '(13824/3125*x+142848/15625)/(x^5+47/15*x^4+4418/1125*x^3-21554/3375*x^2+24529/10125*x-8693/30375)*y+(-13824/3125*x^2+64512/78125*x-3896832/390625)/(x^5+47/15*x^4+4418/1125*x^3-21554/3375*x^2+24529/10125*x-8693/30375)', 'orbit_size': 1, 'pass_size': 2, 'plabel': '5T4-5_5_3.1.1', 'plane_constant': '16/3', 'plane_map_constant_factored': '\\frac{3}{2^{4}} ', 'plane_model': '(-1)*t^2 - (-5*x^2 + 5*x + 4)*t + 9*x^5', 'plane_model_latex': '\\left(-1\\right) t^{2} - \\left(-5 x^{2} + 5 x + 4\\right) t + 9 x^{5}', 'primitivization': '5T4-5_5_3.1.1-a', 'triples': [[[2, 3, 4, 5, 1], [3, 1, 5, 2, 4], [3, 2, 4, 1, 5]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,3,5,4,2)', '(1,3,4)']]}
24. label: 5T4-5_5_3.1.1-b
    {'BelyiDB_label': '5T4-[5,5,3]-5-5-311-g1-b', 'BelyiDB_plabel': '5T4-[5,5,3]-5-5-311-g1', 'a_s': 3, 'abc': [5, 5, 3], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^3+1/375*x+79/337500', 'curve_label': '75.a2', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/75/a/2'], 'g': 1, 'geomtype': 'H', 'group': '5T4', 'group_num': 4, 'is_primitive': True, 'label': '5T4-5_5_3.1.1-b', 'lambdas': [[5], [5], [3, 1, 1]], 'map': '(-27/3125*x-36/15625)/(x^5-7/15*x^4+98/1125*x^3+772/84375*x^2+1672/6328125*x+5792/2373046875)*y+(27/3125*x^2-9/156250*x-318/9765625)/(x^5-7/15*x^4+98/1125*x^3+772/84375*x^2+1672/6328125*x+5792/2373046875)', 'orbit_size': 1, 'pass_size': 2, 'plabel': '5T4-5_5_3.1.1', 'plane_constant': '1/3', 'plane_map_constant_factored': '3', 'plane_model': '(-1)*t^2 + (-10*x^2 + 5*x - 1)*t - 9*x^5', 'plane_model_latex': '\\left(-1\\right) t^{2} + \\left(-10 x^{2} + 5 x - 1\\right) t - 9 x^{5}', 'primitivization': '5T4-5_5_3.1.1-b', 'triples': [[[2, 3, 4, 5, 1], [5, 1, 4, 2, 3], [1, 2, 4, 5, 3]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,5,3,4,2)', '(3,4,5)']]}
25. label: 5T4-5_5_5-a
    {'BelyiDB_label': '5T4-[5,5,5]-5-5-5-g2-a', 'BelyiDB_plabel': '5T4-[5,5,5]-5-5-5-g2', 'a_s': 5, 'abc': [5, 5, 5], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^6+567/512*x^4+111537/1048576*x^2+531441/33554432', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 2, 'geomtype': 'H', 'group': '5T4', 'group_num': 4, 'is_primitive': True, 'label': '5T4-5_5_5-a', 'lambdas': [[5], [5], [5]], 'map': '(x^2+243/1024)/(x^5-45/32*x^4+405/512*x^3-3645/16384*x^2+32805/1048576*x-59049/33554432)*y+(x^5+405/512*x^3+32805/1048576*x+59049/2097152)/(x^5-45/32*x^4+405/512*x^3-3645/16384*x^2+32805/1048576*x-59049/33554432)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T4-5_5_5', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 'x^5*t^3 + (-5*x + 2)*t^2 + (-5*x + 6)*t + 4', 'plane_model_latex': 'x^{5} t^{3} + \\left(-5 x + 2\\right) t^{2} + \\left(-5 x + 6\\right) t + 4', 'primitivization': '5T4-5_5_5-a', 'triples': [[[2, 3, 4, 5, 1], [3, 5, 4, 2, 1], [2, 5, 4, 1, 3]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,3,4,2,5)', '(1,2,5,3,4)']]}
26. label: 5T5-3.2_3.2_2.2.1-a
    {'BelyiDB_label': '5T5-[6,6,2]-32-32-221-g0-a', 'BelyiDB_plabel': '5T5-[6,6,2]-32-32-221-g0', 'a_s': 2, 'abc': [6, 6, 2], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-3.2_3.2_2.2.1-a', 'lambdas': [[3, 2], [3, 2], [2, 2, 1]], 'map': '(156250/165997*x^5-62500/12769*x^4+81250/12769*x^3)/(x^5-485/113*x^4+98215/12769*x^3-36335/12769*x^2-87880/12769*x+114244/12769)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T5-3.2_3.2_2.2.1', 'plane_constant': '1/12500', 'plane_map_constant_factored': '2^{2} \\cdot 5^{5}', 'plane_model': '(-1)*(-25*x + 1)^2*t + x^3*(16*x - 1)^2', 'plane_model_latex': '\\left(-1\\right) \\left(-25 x + 1\\right)^{2} t + x^{3} \\left(16 x - 1\\right)^{2}', 'primitivization': '5T5-3.2_3.2_2.2.1-a', 'triples': [[[2, 3, 1, 5, 4], [5, 1, 4, 3, 2], [4, 2, 5, 1, 3]]], 'triples_cyc': [['(1,2,3)(4,5)', '(1,5,2)(3,4)', '(1,4)(3,5)']]}
27. label: 5T5-3.2_3.2_3.1.1-a
    {'BelyiDB_label': '5T5-[6,6,3]-32-32-311-g0-a', 'BelyiDB_plabel': '5T5-[6,6,3]-32-32-311-g0', 'a_s': 3, 'abc': [6, 6, 3], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-3.2_3.2_3.1.1-a', 'lambdas': [[3, 2], [3, 2], [3, 1, 1]], 'map': '(78125/41261*x^5-31250/41261*x^4+3125/41261*x^3)/(x^5-290/341*x^4+815/3751*x^3-45/41261*x^2-270/41261*x+27/41261)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T5-3.2_3.2_3.1.1', 'plane_constant': '1/3125', 'plane_map_constant_factored': '5^{5}', 'plane_model': '(-25*x + 1)^2*t + x^3*(x - 1)^2', 'plane_model_latex': '\\left(-25 x + 1\\right)^{2} t + x^{3} \\left(x - 1\\right)^{2}', 'primitivization': '5T5-3.2_3.2_3.1.1-a', 'triples': [[[2, 3, 1, 5, 4], [4, 3, 2, 5, 1], [2, 5, 3, 4, 1]]], 'triples_cyc': [['(1,2,3)(4,5)', '(1,4,5)(2,3)', '(1,2,5)']]}
28. label: 5T5-4.1_2.2.1_3.2-a
    {'BelyiDB_label': '5T5-[4,2,6]-41-221-32-g0-a', 'BelyiDB_plabel': '5T5-[4,2,6]-41-221-32-g0', 'a_s': 2, 'abc': [4, 2, 6], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-4.1_2.2.1_3.2-a', 'lambdas': [[4, 1], [2, 2, 1], [3, 2]], 'map': '(140625/140608*x^5-28125/35152*x^4)/(x^5-10/13*x^4-335/169*x^3+3420/2197*x^2+2160/2197*x-1728/2197)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T5-4.1_2.2.1_3.2', 'plane_constant': '27/12500', 'plane_map_constant_factored': '\\frac{2^{2} \\cdot 5^{5}}{3^{3}} ', 'plane_model': '(-1)*(-25*x + 2)*t - x^3*(x + 1)^2', 'plane_model_latex': '\\left(-1\\right) \\left(-25 x + 2\\right) t - x^{3} \\left(x + 1\\right)^{2}', 'primitivization': '5T5-4.1_2.2.1_3.2-a', 'triples': [[[2, 4, 1, 3, 5], [5, 3, 2, 4, 1], [2, 5, 4, 3, 1]]], 'triples_cyc': [['(1,2,4,3)', '(1,5)(2,3)', '(1,2,5)(3,4)']]}
29. label: 5T5-4.1_3.2_3.1.1-a
    {'BelyiDB_label': '5T5-[4,6,3]-41-32-311-g0-a', 'BelyiDB_plabel': '5T5-[4,6,3]-41-32-311-g0', 'a_s': 3, 'abc': [4, 6, 3], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 4, 'base_field': [-6, 0, 1], 'base_field_label': '2.2.24.1', 'c_s': 6, 'curve': 'PP1', 'deg': 5, 'embeddings': [[2.449489742783178, 0.0], [-2.449489742783178, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-4.1_3.2_3.1.1-a', 'lambdas': [[4, 1], [3, 2], [3, 1, 1]], 'map': '(1/313*(-2784*nu+7169)*x^5+1/626*(153*nu-1448)*x^4)/(x^5+1/626*(-287*nu-1048)*x^4+1/313*(-128*nu-282)*x^3+1/2817*(3343*nu+8172)*x^2+1/2817*(5956*nu+14589)*x+1/5634*(-18283*nu-44784))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '5T5-4.1_3.2_3.1.1', 'plane_constant': '1/9*(248*nu - 468)', 'plane_map_constant_factored': '\\frac{3}{2^{2} \\cdot 5^{5}} (62 \\nu + 117)', 'plane_model': '(x - 1)^2*t - x^4*(-2*x - 2*nu + 7)', 'plane_model_latex': '\\left(x - 1\\right)^{2} t - x^{4} \\left(-2 x - 2 \\nu + 7\\right)', 'primitivization': '5T5-4.1_3.2_3.1.1-a', 'triples': [[[2, 4, 3, 5, 1], [3, 5, 1, 2, 4], [2, 3, 1, 4, 5]], [[4, 2, 5, 3, 1], [4, 5, 2, 1, 3], [2, 3, 1, 4, 5]]], 'triples_cyc': [['(1,2,4,5)', '(1,3)(2,5,4)', '(1,2,3)'], ['(1,4,3,5)', '(1,4)(2,5,3)', '(1,2,3)']]}
30. label: 5T5-4.1_4.1_3.1.1-a
    {'BelyiDB_label': '5T5-[4,4,3]-41-41-311-g0-a', 'BelyiDB_plabel': '5T5-[4,4,3]-41-41-311-g0', 'a_s': 3, 'abc': [4, 4, 3], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-4.1_4.1_3.1.1-a', 'lambdas': [[4, 1], [4, 1], [3, 1, 1]], 'map': '(140625/142417*x^5-787500/142417*x^4)/(x^5-18340/3531*x^4+125440/38841*x^3+1756160/142417*x^2+3073280/427251*x-17210368/427251)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T5-4.1_4.1_3.1.1', 'plane_constant': '1/84375', 'plane_map_constant_factored': '3^{3} \\cdot 5^{5}', 'plane_model': '(-1)*(-25*x + 1)*t + x^4*(-9*x + 1)', 'plane_model_latex': '\\left(-1\\right) \\left(-25 x + 1\\right) t + x^{4} \\left(-9 x + 1\\right)', 'primitivization': '5T5-4.1_4.1_3.1.1-a', 'triples': [[[3, 2, 4, 5, 1], [1, 5, 2, 3, 4], [2, 3, 1, 4, 5]]], 'triples_cyc': [['(1,3,4,5)', '(2,5,4,3)', '(1,2,3)']]}
31. label: 5T5-5_2.1.1.1_3.2-a
    {'BelyiDB_label': '5T5-[5,2,6]-5-2111-32-g0-a', 'BelyiDB_plabel': '5T5-[5,2,6]-5-2111-32-g0', 'a_s': 2, 'abc': [5, 2, 6], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-5_2.1.1.1_3.2-a', 'lambdas': [[5], [2, 1, 1, 1], [3, 2]], 'map': '3/2*x^5/(x^5-5/2*x^4-5/4*x^3+45/8*x^2-27/8)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T5-5_2.1.1.1_3.2', 'plane_constant': '108/3125', 'plane_map_constant_factored': '\\frac{5^{5}}{2^{2} \\cdot 3^{3}} ', 'plane_model': '(-1)*x^5*t + (x - 1)^2', 'plane_model_latex': '\\left(-1\\right) x^{5} t + \\left(x - 1\\right)^{2}', 'primitivization': '5T5-5_2.1.1.1_3.2-a', 'triples': [[[3, 5, 2, 1, 4], [2, 1, 3, 4, 5], [4, 3, 2, 5, 1]]], 'triples_cyc': [['(1,3,2,5,4)', '(1,2)', '(1,4,5)(2,3)']]}
32. label: 5T5-5_4.1_2.1.1.1-a
    {'BelyiDB_label': '5T5-[5,4,2]-5-41-2111-g0-a', 'BelyiDB_plabel': '5T5-[5,4,2]-5-41-2111-g0', 'a_s': 2, 'abc': [5, 4, 2], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-5_4.1_2.1.1.1-a', 'lambdas': [[5], [4, 1], [2, 1, 1, 1]], 'map': '64/63*x^5/(x^5-85/504*x^4-1445/2268*x^3-24565/27216*x^2+1419857/1959552)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T5-5_4.1_2.1.1.1', 'plane_constant': '1/256', 'plane_map_constant_factored': '2^{8}', 'plane_model': '(-1)*(-5*x + 1)*t + x^5', 'plane_model_latex': '\\left(-1\\right) \\left(-5 x + 1\\right) t + x^{5}', 'primitivization': '5T5-5_4.1_2.1.1.1-a', 'triples': [[[2, 3, 4, 5, 1], [1, 5, 2, 3, 4], [2, 1, 3, 4, 5]]], 'triples_cyc': [['(1,2,3,4,5)', '(2,5,4,3)', '(1,2)']]}
33. label: 5T5-5_3.2_3.2-a
    {'BelyiDB_label': '5T5-[5,6,6]-5-32-32-g1-a', 'BelyiDB_plabel': '5T5-[5,6,6]-5-32-32-g1', 'a_s': 5, 'abc': [5, 6, 6], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^3-1953125/459165024*x+45166015625/289207845356544', 'curve_label': '900.e2', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/900/e/2'], 'g': 1, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-5_3.2_3.2-a', 'lambdas': [[5], [3, 2], [3, 2]], 'map': '(-9765625/11019960576*x-30517578125/1156831381426176)/(x^5+3125/52488*x^4-37109375/11019960576*x^3-28076171875/289207845356544*x^2+41961669921875/7589970693537140736*x-11539459228515625/199191190881188721475584)*y-95367431640625/242879062193188503552/(x^5+3125/52488*x^4-37109375/11019960576*x^3-28076171875/289207845356544*x^2+41961669921875/7589970693537140736*x-11539459228515625/199191190881188721475584)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T5-5_3.2_3.2', 'plane_constant': '12', 'plane_map_constant_factored': '\\frac{1}{2^{2} \\cdot 3} ', 'plane_model': '(-3)*t^2 + 36*t - x^3*(x - 5)^2', 'plane_model_latex': '\\left(-3\\right) t^{2} + 36 t - x^{3} \\left(x - 5\\right)^{2}', 'primitivization': '5T5-5_3.2_3.2-a', 'triples': [[[4, 5, 1, 2, 3], [2, 3, 1, 5, 4], [2, 5, 4, 3, 1]]], 'triples_cyc': [['(1,4,2,5,3)', '(1,2,3)(4,5)', '(1,2,5)(3,4)']]}
34. label: 5T5-5_3.2_4.1-a
    {'BelyiDB_label': '5T5-[5,6,4]-5-32-41-g1-a', 'BelyiDB_plabel': '5T5-[5,6,4]-5-32-41-g1', 'a_s': 4, 'abc': [5, 6, 4], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 5, 'base_field': [-6, 0, 1], 'base_field_label': '2.2.24.1', 'c_s': 6, 'curve': 'y^2=x^3+1/146484375*(-5655936*nu-13761101)*x+1/20599365234375*(-32784398208*nu-79530479078)', 'deg': 5, 'embeddings': [[-2.449489742783178, 0.0], [2.449489742783178, 0.0]], 'friends': [], 'g': 1, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-5_3.2_4.1-a', 'lambdas': [[5], [3, 2], [4, 1]], 'map': '(1/30517578125*(1069645824*nu+2709725184)*x+1/95367431640625*(92066512896*nu+290402574336))/(x^5+1/9375*(-768*nu-6913)*x^4+1/439453125*(-25294848*nu-35320318)*x^3+1/20599365234375*(-63299962368*nu-213845723138)*x^2+1/193119049072265625*(-27296160448512*nu-54974447094067)*x+1/226311385631561279296875*(-358703942993689344*nu-992752349979472129))*y+(1/30517578125*(-1069645824*nu-2709725184)*x^2+1/476837158203125*(-288231063552*nu-701587832832)*x+1/931322574615478515625*(31029244749840384*nu+71661730221318144))/(x^5+1/9375*(-768*nu-6913)*x^4+1/439453125*(-25294848*nu-35320318)*x^3+1/20599365234375*(-63299962368*nu-213845723138)*x^2+1/193119049072265625*(-27296160448512*nu-54974447094067)*x+1/226311385631561279296875*(-358703942993689344*nu-992752349979472129))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '5T5-5_3.2_4.1', 'plane_constant': '1/9*(80*nu + 196)', 'plane_map_constant_factored': '\\frac{3^{2}}{2^{2}} (-20 \\nu + 49)', 'plane_model': '(-4)*t^2 - (2*nu - 5)*(15*x^3 + (-10*nu - 20)*x^2 + (10*nu + 25)*x - 356*nu - 872)*t + 3*x^5', 'plane_model_latex': '\\left(-4\\right) t^{2} - \\left(2 \\nu - 5\\right) \\left(15 x^{3} + \\left(-10 \\nu - 20\\right) x^{2} + \\left(10 \\nu + 25\\right) x - 356 \\nu - 872\\right) t + 3 x^{5}', 'primitivization': '5T5-5_3.2_4.1-a', 'triples': [[[2, 3, 4, 5, 1], [2, 3, 1, 5, 4], [4, 3, 1, 2, 5]], [[2, 3, 4, 5, 1], [3, 5, 1, 2, 4], [2, 3, 4, 1, 5]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,2,3)(4,5)', '(1,4,2,3)'], ['(1,2,3,4,5)', '(1,3)(2,5,4)', '(1,2,3,4)']]}
35. label: 5T5-5_4.1_4.1-a
    {'BelyiDB_label': '5T5-[5,4,4]-5-41-41-g1-a', 'BelyiDB_plabel': '5T5-[5,4,4]-5-41-41-g1', 'a_s': 4, 'abc': [5, 4, 4], 'aut_group': [[1, 2, 3, 4, 5]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^3+1953125/1048576*x+1220703125/536870912', 'curve_label': '400.a1', 'deg': 5, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/400/a/1'], 'g': 1, 'geomtype': 'H', 'group': '5T5', 'group_num': 5, 'is_primitive': True, 'label': '5T5-5_4.1_4.1-a', 'lambdas': [[5], [4, 1], [4, 1]], 'map': '(-9765625/1048576*x+30517578125/1073741824)/(x^5-3125/512*x^4+5859375/524288*x^3-1220703125/134217728*x^2+3814697265625/1099511627776*x-286102294921875/562949953421312)*y-95367431640625/2199023255552/(x^5-3125/512*x^4+5859375/524288*x^3-1220703125/134217728*x^2+3814697265625/1099511627776*x-286102294921875/562949953421312)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '5T5-5_4.1_4.1', 'plane_constant': '2', 'plane_map_constant_factored': '\\frac{1}{2} ', 'plane_model': '(-1)*t^2 + 2*t - x^4*(4*x + 5)', 'plane_model_latex': '\\left(-1\\right) t^{2} + 2 t - x^{4} \\left(4 x + 5\\right)', 'primitivization': '5T5-5_4.1_4.1-a', 'triples': [[[4, 5, 1, 2, 3], [2, 3, 4, 1, 5], [2, 3, 5, 4, 1]]], 'triples_cyc': [['(1,4,2,5,3)', '(1,2,3,4)', '(1,2,3,5)']]}
36. label: 6T1-6_6_1.1.1.1.1.1-a
    {'BelyiDB_label': '6T1-[6,6,1]-6-6-111111-g0-a', 'BelyiDB_plabel': '6T1-[6,6,1]-6-6-111111-g0', 'a_s': 1, 'abc': [6, 6, 1], 'aut_group': [[2, 3, 4, 5, 6, 1]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '6T1', 'group_num': 1, 'is_primitive': False, 'label': '6T1-6_6_1.1.1.1.1.1-a', 'lambdas': [[6], [6], [1, 1, 1, 1, 1, 1]], 'map': '-1/(x^6-1)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T1-6_6_1.1.1.1.1.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't + x^6', 'plane_model_latex': 't + x^{6}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[2, 3, 4, 5, 6, 1], [6, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6]]], 'triples_cyc': [['(1,2,3,4,5,6)', '(1,6,5,4,3,2)', '()']]}
37. label: 6T1-6_2.2.2_3.3-a
    {'BelyiDB_label': '6T1-[6,2,3]-6-222-33-g1-a', 'BelyiDB_plabel': '6T1-[6,2,3]-6-222-33-g1', 'a_s': 2, 'abc': [6, 2, 3], 'aut_group': [[5, 6, 1, 2, 3, 4], [6, 1, 2, 3, 4, 5]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^3-1', 'curve_label': '144.a3', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/144/a/3'], 'g': 1, 'geomtype': 'E', 'group': '6T1', 'group_num': 1, 'is_primitive': False, 'label': '6T1-6_2.2.2_3.3-a', 'lambdas': [[6], [2, 2, 2], [3, 3]], 'map': '1/x^3', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T1-6_2.2.2_3.3', 'plane_constant': '1/9', 'plane_map_constant_factored': '3^{2}', 'plane_model': '9*x^6*t^2 + (-3*x^4 + 6*x^2 - 1)*t + x^2', 'plane_model_latex': '9 x^{6} t^{2} + \\left(-3 x^{4} + 6 x^{2} - 1\\right) t + x^{2}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[2, 3, 4, 5, 6, 1], [4, 5, 6, 1, 2, 3], [3, 4, 5, 6, 1, 2]]], 'triples_cyc': [['(1,2,3,4,5,6)', '(1,4)(2,5)(3,6)', '(1,3,5)(2,4,6)']]}
38. label: 6T1-6_6_3.3-a
    {'BelyiDB_label': '6T1-[6,6,3]-6-6-33-g2-a', 'BelyiDB_plabel': '6T1-[6,6,3]-6-6-33-g2', 'a_s': 3, 'abc': [6, 6, 3], 'aut_group': [[2, 3, 4, 5, 6, 1]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^6-2', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 2, 'geomtype': 'H', 'group': '6T1', 'group_num': 1, 'is_primitive': False, 'label': '6T1-6_6_3.3-a', 'lambdas': [[6], [6], [3, 3]], 'map': '1/2/x^3*y+1/2', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T1-6_6_3.3', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '2*t^2 - 2*t + x^6', 'plane_model_latex': '2 t^{2} - 2 t + x^{6}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[2, 3, 4, 5, 6, 1], [2, 3, 4, 5, 6, 1], [5, 6, 1, 2, 3, 4]]], 'triples_cyc': [['(1,2,3,4,5,6)', '(1,2,3,4,5,6)', '(1,5,3)(2,6,4)']]}
39. label: 6T2-3.3_2.2.2_2.2.2-a
    {'BelyiDB_label': '6T2-[3,2,2]-33-222-222-g0-a', 'BelyiDB_plabel': '6T2-[3,2,2]-33-222-222-g0', 'a_s': 2, 'abc': [3, 2, 2], 'aut_group': [[5, 4, 1, 6, 3, 2], [4, 5, 6, 1, 2, 3]], 'b_s': 2, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 3, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '6T2', 'group_num': 2, 'is_primitive': False, 'label': '6T2-3.3_2.2.2_2.2.2-a', 'lambdas': [[3, 3], [2, 2, 2], [2, 2, 2]], 'map': '-4*x^3/(x^6-2*x^3+1)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T2-3.3_2.2.2_2.2.2', 'plane_constant': '4', 'plane_map_constant_factored': '\\frac{1}{2^{2}} ', 'plane_model': 'x^3*t + (x + 1)^2*(x^2 - x + 1)^2', 'plane_model_latex': 'x^{3} t + \\left(x + 1\\right)^{2} \\left(x^{2} - x + 1\\right)^{2}', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[3, 4, 5, 6, 1, 2], [2, 1, 6, 5, 4, 3], [4, 3, 2, 1, 6, 5]]], 'triples_cyc': [['(1,3,5)(2,4,6)', '(1,2)(3,6)(4,5)', '(1,4)(2,3)(5,6)']]}
40. label: 6T3-6_2.2.2_2.2.1.1-a
    {'BelyiDB_label': '6T3-[6,2,2]-6-222-2211-g0-a', 'BelyiDB_plabel': '6T3-[6,2,2]-6-222-2211-g0', 'a_s': 2, 'abc': [6, 2, 2], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 2, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '6T3', 'group_num': 3, 'is_primitive': False, 'label': '6T3-6_2.2.2_2.2.1.1-a', 'lambdas': [[6], [2, 2, 2], [2, 2, 1, 1]], 'map': '-1/16/(x^6-3/2*x^4+9/16*x^2-1/16)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T3-6_2.2.2_2.2.1.1', 'plane_constant': '4', 'plane_map_constant_factored': '\\frac{1}{2^{2}} ', 'plane_model': 't + (x - 2)*(x - 1)^2*(x + 1)^2*(x + 2)', 'plane_model_latex': 't + \\left(x - 2\\right) \\left(x - 1\\right)^{2} \\left(x + 1\\right)^{2} \\left(x + 2\\right)', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[2, 3, 4, 5, 6, 1], [4, 3, 2, 1, 6, 5], [5, 4, 3, 2, 1, 6]]], 'triples_cyc': [['(1,2,3,4,5,6)', '(1,4)(2,3)(5,6)', '(1,5)(2,4)']]}
41. label: 6T4-3.3_3.3_2.2.1.1-a
    {'BelyiDB_label': '6T4-[3,3,2]-33-33-2211-g0-a', 'BelyiDB_plabel': '6T4-[3,3,2]-33-33-2211-g0', 'a_s': 2, 'abc': [3, 3, 2], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 3, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '6T4', 'group_num': 4, 'is_primitive': False, 'label': '6T4-3.3_3.3_2.2.1.1-a', 'lambdas': [[3, 3], [3, 3], [2, 2, 1, 1]], 'map': '(-357911/2566296*x^6+357911/285144*x^4-357911/95048*x^2+357911/95048)/(x^6-20449/1308*x^5+8373931/95048*x^4-27182/109*x^3+36701325/95048*x^2-34347/109*x+855/8)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T4-3.3_3.3_2.2.1.1', 'plane_constant': '36', 'plane_map_constant_factored': '\\frac{1}{2^{2} \\cdot 3^{2}} ', 'plane_model': 'x*(x^2 + 3)^2*t + (x^2 - 6*x - 3)^3', 'plane_model_latex': 'x \\left(x^{2} + 3\\right)^{2} t + \\left(x^{2} - 6 x - 3\\right)^{3}', 'primitivization': '3T1-3_3_1.1.1-a', 'triples': [[[3, 4, 5, 6, 1, 2], [2, 6, 4, 5, 3, 1], [4, 2, 6, 1, 5, 3]]], 'triples_cyc': [['(1,3,5)(2,4,6)', '(1,2,6)(3,4,5)', '(1,4)(3,6)']]}
42. label: 6T4-3.3_3.3_3.3-a
    {'BelyiDB_label': '6T4-[3,3,3]-33-33-33-g1-a', 'BelyiDB_plabel': '6T4-[3,3,3]-33-33-33-g1', 'a_s': 3, 'abc': [3, 3, 3], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 3, 'curve': 'y^2=x^3-15*x+22', 'curve_label': '36.a2', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/36/a/2'], 'g': 1, 'geomtype': 'E', 'group': '6T4', 'group_num': 4, 'is_primitive': False, 'label': '6T4-3.3_3.3_3.3-a', 'lambdas': [[3, 3], [3, 3], [3, 3]], 'map': '(-1/16*x^2+1/4*x-7/16)/(x^2-4*x+4)*y+1/2', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T4-3.3_3.3_3.3', 'plane_constant': '1/16', 'plane_map_constant_factored': '2^{4}', 'plane_model': '(-16)*t^2 + t + x^3*(x - 1)^3', 'plane_model_latex': '\\left(-16\\right) t^{2} + t + x^{3} \\left(x - 1\\right)^{3}', 'primitivization': '3T1-3_3_3-a', 'triples': [[[3, 4, 5, 6, 1, 2], [6, 4, 2, 3, 1, 5], [6, 1, 5, 3, 4, 2]]], 'triples_cyc': [['(1,3,5)(2,4,6)', '(1,6,5)(2,4,3)', '(1,6,2)(3,5,4)']]}
43. label: 6T5-6_2.2.2_3.1.1.1-a
    {'BelyiDB_label': '6T5-[6,2,3]-6-222-3111-g0-a', 'BelyiDB_plabel': '6T5-[6,2,3]-6-222-3111-g0', 'a_s': 2, 'abc': [6, 2, 3], 'aut_group': [[5, 6, 1, 2, 3, 4]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'E', 'group': '6T5', 'group_num': 5, 'is_primitive': False, 'label': '6T5-6_2.2.2_3.1.1.1-a', 'lambdas': [[6], [2, 2, 2], [3, 1, 1, 1]], 'map': '-1/4/(x^6+x^3)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T5-6_2.2.2_3.1.1.1', 'plane_constant': '1/4', 'plane_map_constant_factored': '2^{2}', 'plane_model': 't + x^3*(x + 1)*(x^2 - x + 1)', 'plane_model_latex': 't + x^{3} \\left(x + 1\\right) \\left(x^{2} - x + 1\\right)', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[4, 3, 6, 5, 2, 1], [4, 5, 6, 1, 2, 3], [3, 2, 5, 4, 1, 6]]], 'triples_cyc': [['(1,4,5,2,3,6)', '(1,4)(2,5)(3,6)', '(1,3,5)']]}
44. label: 6T5-6_6_3.1.1.1-a
    {'BelyiDB_label': '6T5-[6,6,3]-6-6-3111-g1-a', 'BelyiDB_plabel': '6T5-[6,6,3]-6-6-3111-g1', 'a_s': 3, 'abc': [6, 6, 3], 'aut_group': [[5, 6, 1, 2, 3, 4]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^3+1/4', 'curve_label': '27.a4', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/27/a/4'], 'g': 1, 'geomtype': 'H', 'group': '6T5', 'group_num': 5, 'is_primitive': False, 'label': '6T5-6_6_3.1.1.1-a', 'lambdas': [[6], [6], [3, 1, 1, 1]], 'map': '1/(x^6-2*x^3)*y+(-x^3+1/2)/(x^6-2*x^3)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T5-6_6_3.1.1.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't^2 - (2*x^3 - 1)*t + x^6', 'plane_model_latex': 't^{2} - \\left(2 x^{3} - 1\\right) t + x^{6}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[6, 5, 2, 1, 4, 3], [2, 3, 4, 5, 6, 1], [3, 2, 5, 4, 1, 6]]], 'triples_cyc': [['(1,6,3,2,5,4)', '(1,2,3,4,5,6)', '(1,3,5)']]}
45. label: 6T5-6_6_3.3-a
    {'BelyiDB_label': '6T5-[6,6,3]-6-6-33-g2-a', 'BelyiDB_plabel': '6T5-[6,6,3]-6-6-33-g2', 'a_s': 3, 'abc': [6, 6, 3], 'aut_group': [[5, 6, 1, 2, 3, 4]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^6-3*x^3+1/4', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 2, 'geomtype': 'H', 'group': '6T5', 'group_num': 5, 'is_primitive': False, 'label': '6T5-6_6_3.3-a', 'lambdas': [[6], [6], [3, 3]], 'map': '(-1/4*x^3+1/8)/x^3*y+(-1/4*x^6+1/2*x^3+1/16)/x^3', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T5-6_6_3.3', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '4*t^2 + (x^6 - 8*x^3 + 4)*t + x^6', 'plane_model_latex': '4 t^{2} + \\left(x^{6} - 8 x^{3} + 4\\right) t + x^{6}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[6, 5, 2, 1, 4, 3], [6, 1, 2, 3, 4, 5], [5, 4, 1, 6, 3, 2]]], 'triples_cyc': [['(1,6,3,2,5,4)', '(1,6,5,4,3,2)', '(1,5,3)(2,4,6)']]}
46. label: 6T6-6_3.3_2.1.1.1.1-a
    {'BelyiDB_label': '6T6-[6,3,2]-6-33-21111-g0-a', 'BelyiDB_plabel': '6T6-[6,3,2]-6-33-21111-g0', 'a_s': 2, 'abc': [6, 3, 2], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'E', 'group': '6T6', 'group_num': 6, 'is_primitive': False, 'label': '6T6-6_3.3_2.1.1.1.1-a', 'lambdas': [[6], [3, 3], [2, 1, 1, 1, 1]], 'map': '1/27/(x^6-x^4+1/3*x^2)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T6-6_3.3_2.1.1.1.1', 'plane_constant': '1/9', 'plane_map_constant_factored': '3^{2}', 'plane_model': 'x^6*t - (x^4 - 3*x^2 + 3)', 'plane_model_latex': 'x^{6} t - \\left(x^{4} - 3 x^{2} + 3\\right)', 'primitivization': '3T1-3_3_1.1.1-a', 'triples': [[[6, 4, 5, 3, 1, 2], [2, 6, 4, 5, 3, 1], [4, 2, 3, 1, 5, 6]]], 'triples_cyc': [['(1,6,2,4,3,5)', '(1,2,6)(3,4,5)', '(1,4)']]}
47. label: 6T6-6_6_2.2.1.1-a
    {'BelyiDB_label': '6T6-[6,6,2]-6-6-2211-g1-a', 'BelyiDB_plabel': '6T6-[6,6,2]-6-6-2211-g1', 'a_s': 2, 'abc': [6, 6, 2], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^3+2/27*x-7/729', 'curve_label': '72.a5', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/72/a/5'], 'g': 1, 'geomtype': 'H', 'group': '6T6', 'group_num': 6, 'is_primitive': False, 'label': '6T6-6_6_2.2.1.1-a', 'lambdas': [[6], [6], [2, 2, 1, 1]], 'map': '-1/27/(x^3-1/3*x^2+1/27*x-28/729)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T6-6_6_2.2.1.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't^2 + (-3*x^2 + 1)*(-9*x^4 + 6*x^2 + 2)*t + 1', 'plane_model_latex': 't^{2} + \\left(-3 x^{2} + 1\\right) \\left(-9 x^{4} + 6 x^{2} + 2\\right) t + 1', 'primitivization': '3T1-3_3_1.1.1-a', 'triples': [[[6, 4, 5, 3, 1, 2], [2, 3, 4, 5, 6, 1], [4, 5, 3, 1, 2, 6]]], 'triples_cyc': [['(1,6,2,4,3,5)', '(1,2,3,4,5,6)', '(1,4)(2,5)']]}
48. label: 6T6-6_6_3.3-a
    {'BelyiDB_label': '6T6-[6,6,3]-6-6-33-g2-a', 'BelyiDB_plabel': '6T6-[6,6,3]-6-6-33-g2', 'a_s': 3, 'abc': [6, 6, 3], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^6+4*x^4+6*x^2+3', 'curve_label': '1728.b.442368.1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['Genus2Curve/Q/1728/b/442368/1'], 'g': 2, 'geomtype': 'H', 'group': '6T6', 'group_num': 6, 'is_primitive': False, 'label': '6T6-6_6_3.3-a', 'lambdas': [[6], [6], [3, 3]], 'map': '1/2*x/(x^4+2*x^2+1)*y+1/2', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T6-6_6_3.3', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '(-1)*t^3 - (-3*x^2 + 4)*t^2 + (-3*x^4 - 4)*t + x^6', 'plane_model_latex': '\\left(-1\\right) t^{3} - \\left(-3 x^{2} + 4\\right) t^{2} + \\left(-3 x^{4} - 4\\right) t + x^{6}', 'primitivization': '3T1-3_3_3-a', 'triples': [[[6, 4, 5, 3, 1, 2], [3, 1, 5, 6, 4, 2], [3, 4, 5, 6, 1, 2]]], 'triples_cyc': [['(1,6,2,4,3,5)', '(1,3,5,4,6,2)', '(1,3,5)(2,4,6)']]}
49. label: 6T7-4.2_3.3_2.2.1.1-a
    {'BelyiDB_label': '6T7-[4,3,2]-42-33-2211-g0-a', 'BelyiDB_plabel': '6T7-[4,3,2]-42-33-2211-g0', 'a_s': 2, 'abc': [4, 3, 2], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '6T7', 'group_num': 7, 'is_primitive': False, 'label': '6T7-4.2_3.3_2.2.1.1-a', 'lambdas': [[4, 2], [3, 3], [2, 2, 1, 1]], 'map': '-108*x^2/(x^6+12*x^4-60*x^2+64)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T7-4.2_3.3_2.2.1.1', 'plane_constant': '27/4', 'plane_map_constant_factored': '\\frac{2^{2}}{3^{3}} ', 'plane_model': 'x^2*t + (x^2 + 1)^3', 'plane_model_latex': 'x^{2} t + \\left(x^{2} + 1\\right)^{3}', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[3, 5, 4, 6, 2, 1], [2, 3, 1, 5, 6, 4], [5, 4, 3, 2, 1, 6]]], 'triples_cyc': [['(1,3,4,6)(2,5)', '(1,2,3)(4,5,6)', '(1,5)(2,4)']]}
50. label: 6T7-4.2_4.2_3.3-a
    {'BelyiDB_label': '6T7-[4,4,3]-42-42-33-g1-a', 'BelyiDB_plabel': '6T7-[4,4,3]-42-42-33-g1', 'a_s': 3, 'abc': [4, 4, 3], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'y^2=x^3-73/107163*x+170/26040609', 'curve_label': '7056.q3', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/7056/q/3'], 'g': 1, 'geomtype': 'H', 'group': '6T7', 'group_num': 7, 'is_primitive': False, 'label': '6T7-4.2_4.2_3.3-a', 'lambdas': [[4, 2], [4, 2], [3, 3]], 'map': '(1/1323*x-1/107163)/(x^3-1/189*x^2+1/107163*x-1/182284263)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T7-4.2_4.2_3.3', 'plane_constant': '1/147', 'plane_map_constant_factored': '3 \\cdot 7^{2}', 'plane_model': '7*x^6*t^2 + 3*x^2*(x^2 - 21)*t - 3', 'plane_model_latex': '7 x^{6} t^{2} + 3 x^{2} \\left(x^{2} - 21\\right) t - 3', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[5, 1, 6, 2, 4, 3], [3, 5, 4, 6, 2, 1], [5, 3, 4, 2, 6, 1]]], 'triples_cyc': [['(1,5,4,2)(3,6)', '(1,3,4,6)(2,5)', '(1,5,6)(2,3,4)']]}
51. label: 6T8-4.1.1_3.3_2.2.2-a
    {'BelyiDB_label': '6T8-[4,3,2]-411-33-222-g0-a', 'BelyiDB_plabel': '6T8-[4,3,2]-411-33-222-g0', 'a_s': 2, 'abc': [4, 3, 2], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '6T8', 'group_num': 8, 'is_primitive': False, 'label': '6T8-4.1.1_3.3_2.2.2-a', 'lambdas': [[4, 1, 1], [3, 3], [2, 2, 2]], 'map': '(-27/16*x^2+27/8)/(x^6-9/2*x^4+81/16*x^2)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T8-4.1.1_3.3_2.2.2', 'plane_constant': '1/27', 'plane_map_constant_factored': '3^{3}', 'plane_model': '(-1)*(-9*x^2 + 8)*t + x^2*(x - 1)^2*(x + 1)^2', 'plane_model_latex': '\\left(-1\\right) \\left(-9 x^{2} + 8\\right) t + x^{2} \\left(x - 1\\right)^{2} \\left(x + 1\\right)^{2}', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[3, 2, 4, 6, 5, 1], [5, 3, 4, 2, 6, 1], [5, 4, 6, 2, 1, 3]]], 'triples_cyc': [['(1,3,4,6)', '(1,5,6)(2,3,4)', '(1,5)(2,4)(3,6)']]}
52. label: 6T8-4.1.1_4.1.1_3.3-a
    {'BelyiDB_label': '6T8-[4,4,3]-411-411-33-g0-a', 'BelyiDB_plabel': '6T8-[4,4,3]-411-411-33-g0', 'a_s': 3, 'abc': [4, 4, 3], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T8', 'group_num': 8, 'is_primitive': False, 'label': '6T8-4.1.1_4.1.1_3.3-a', 'lambdas': [[4, 1, 1], [4, 1, 1], [3, 3]], 'map': '(x^6+4*x^4)/(x^6+4*x^4+16/3*x^2+64/27)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T8-4.1.1_4.1.1_3.3', 'plane_constant': '1/27', 'plane_map_constant_factored': '3^{3}', 'plane_model': '(-9*x^2 - 1)*t + x^4*(x^2 + 1)', 'plane_model_latex': '\\left(-9 x^{2} - 1\\right) t + x^{4} \\left(x^{2} + 1\\right)', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[5, 1, 3, 2, 4, 6], [3, 2, 4, 6, 5, 1], [2, 3, 1, 5, 6, 4]]], 'triples_cyc': [['(1,5,4,2)', '(1,3,4,6)', '(1,2,3)(4,5,6)']]}
53. label: 6T9-6_6_2.2.1.1-a
    {'BelyiDB_label': '6T9-[6,6,2]-6-6-2211-g1-a', 'BelyiDB_plabel': '6T9-[6,6,2]-6-6-2211-g1', 'a_s': 2, 'abc': [6, 6, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^3+7/432*x-13/23328', 'curve_label': '54.b3', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/54/b/3'], 'g': 1, 'geomtype': 'H', 'group': '6T9', 'group_num': 9, 'is_primitive': False, 'label': '6T9-6_6_2.2.1.1-a', 'lambdas': [[6], [6], [2, 2, 1, 1]], 'map': '(-4/729*x+1/19683)/(x^6-1/2*x^5+5/48*x^4-7/11664*x^3+647/186624*x^2+1037/10077696*x+53465/2176782336)*y+(4/729*x^3+1/729*x^2+5/78732*x+77/2834352)/(x^6-1/2*x^5+5/48*x^4-7/11664*x^3+647/186624*x^2+1037/10077696*x+53465/2176782336)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T9-6_6_2.2.1.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 't^2 + (x - 1)^2*(4*x - 1)*t + 4*x^6', 'plane_model_latex': 't^{2} + \\left(x - 1\\right)^{2} \\left(4 x - 1\\right) t + 4 x^{6}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[6, 5, 2, 1, 4, 3], [2, 5, 6, 3, 4, 1], [5, 4, 3, 2, 1, 6]]], 'triples_cyc': [['(1,6,3,2,5,4)', '(1,2,5,4,3,6)', '(1,5)(2,4)']]}
54. label: 6T10-4.2_4.2_2.2.1.1-a
    {'BelyiDB_label': '6T10-[4,4,2]-42-42-2211-g0-a', 'BelyiDB_plabel': '6T10-[4,4,2]-42-42-2211-g0', 'a_s': 2, 'abc': [4, 4, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-3, 0, 1], 'base_field_label': '2.2.12.1', 'c_s': 4, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.732050807568877, 0.0], [-1.732050807568877, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'E', 'group': '6T10', 'group_num': 10, 'is_primitive': False, 'label': '6T10-4.2_4.2_2.2.1.1-a', 'lambdas': [[4, 2], [4, 2], [2, 2, 1, 1]], 'map': '(x^6-2*x^5+x^4)/(x^6-2*x^5+x^4+1/9*(2*nu+3)*x^2+1/27*(-8*nu-14)*x+1/243*(26*nu+45))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '6T10-4.2_4.2_2.2.1.1', 'plane_constant': '-567*nu + 972', 'plane_map_constant_factored': '\\frac{1}{3^{5}} (-7 \\nu - 12)', 'plane_model': 'x^4*(x - 1)^2*t - (nu - 2)*(-3*nu*x + nu + 2)^2', 'plane_model_latex': 'x^{4} \\left(x - 1\\right)^{2} t - \\left(\\nu - 2\\right) \\left(3 \\nu x - \\nu - 2\\right)^{2}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[4, 1, 6, 5, 2, 3], [6, 1, 2, 5, 4, 3], [3, 4, 1, 2, 5, 6]], [[4, 1, 6, 5, 2, 3], [2, 5, 4, 3, 6, 1], [1, 2, 5, 6, 3, 4]]], 'triples_cyc': [['(1,4,5,2)(3,6)', '(1,6,3,2)(4,5)', '(1,3)(2,4)'], ['(1,4,5,2)(3,6)', '(1,2,5,6)(3,4)', '(3,5)(4,6)']]}
55. label: 6T10-4.2_4.2_3.1.1.1-a
    {'BelyiDB_label': '6T10-[4,4,3]-42-42-3111-g0-a', 'BelyiDB_plabel': '6T10-[4,4,3]-42-42-3111-g0', 'a_s': 3, 'abc': [4, 4, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T10', 'group_num': 10, 'is_primitive': False, 'label': '6T10-4.2_4.2_3.1.1.1-a', 'lambdas': [[4, 2], [4, 2], [3, 1, 1, 1]], 'map': '(6561/161*x^6-74358/161*x^5+210681/161*x^4)/(x^6-9078/161*x^5+182937/161*x^4-1493552/161*x^3+3257319/161*x^2+8519142/161*x-24137569/161)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T10-4.2_4.2_3.1.1.1', 'plane_constant': '1/729', 'plane_map_constant_factored': '3^{6}', 'plane_model': '(-9*x + 1)^2*t + x^4*(x - 1)^2', 'plane_model_latex': '\\left(-9 x + 1\\right)^{2} t + x^{4} \\left(x - 1\\right)^{2}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[4, 1, 6, 5, 2, 3], [2, 3, 6, 5, 4, 1], [1, 4, 3, 6, 5, 2]]], 'triples_cyc': [['(1,4,5,2)(3,6)', '(1,2,3,6)(4,5)', '(2,4,6)']]}
56. label: 6T10-4.2_4.2_3.3-a
    {'BelyiDB_label': '6T10-[4,4,3]-42-42-33-g1-a', 'BelyiDB_plabel': '6T10-[4,4,3]-42-42-33-g1', 'a_s': 3, 'abc': [4, 4, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'y^2=x^3+1007401764096/602425897921*x+273179726831591424/467579487356261281', 'curve_label': '432.b3', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/432/b/3'], 'g': 1, 'geomtype': 'H', 'group': '6T10', 'group_num': 10, 'is_primitive': False, 'label': '6T10-4.2_4.2_3.3-a', 'lambdas': [[4, 2], [4, 2], [3, 3]], 'map': '(1077632058507264/530737216068401*x^2-472054567964989980672/411937528360866188561*x-224014439336921925307858944/319729843950098261779694321)/(x^6-1314144/776161*x^5+3022205292288/602425897921*x^4-2227465464934514688/467579487356261281*x^3+2464655906166208312836096/362916962485923112122241*x^2-873992335073000891588611670016/281681992520036568627910696801*x+542371675391582017291361566838489088/218630576996344103142807794339760961)*y+(1726974452736/602425897921*x^4-1344884809017065472/467579487356261281*x^3+2595828069239479903715328/362916962485923112122241*x^2-1403226448006478940128428425216/281681992520036568627910696801*x+419060223852319425531855995871952896/218630576996344103142807794339760961)/(x^6-1314144/776161*x^5+3022205292288/602425897921*x^4-2227465464934514688/467579487356261281*x^3+2464655906166208312836096/362916962485923112122241*x^2-873992335073000891588611670016/281681992520036568627910696801*x+542371675391582017291361566838489088/218630576996344103142807794339760961)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T10-4.2_4.2_3.3', 'plane_constant': '8', 'plane_map_constant_factored': '\\frac{1}{2^{3}} ', 'plane_model': 't^2 - 8*t + x^4*(x - 3)^2', 'plane_model_latex': 't^{2} - 8 t + x^{4} \\left(x - 3\\right)^{2}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[4, 1, 6, 5, 2, 3], [4, 1, 2, 3, 6, 5], [3, 6, 5, 2, 1, 4]]], 'triples_cyc': [['(1,4,5,2)(3,6)', '(1,4,3,2)(5,6)', '(1,3,5)(2,6,4)']]}
57. label: 6T11-6_2.2.1.1_4.1.1-a
    {'BelyiDB_label': '6T11-[6,2,4]-6-2211-411-g0-a', 'BelyiDB_plabel': '6T11-[6,2,4]-6-2211-411-g0', 'a_s': 2, 'abc': [6, 2, 4], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T11', 'group_num': 11, 'is_primitive': False, 'label': '6T11-6_2.2.1.1_4.1.1-a', 'lambdas': [[6], [2, 2, 1, 1], [4, 1, 1]], 'map': '1/3*x^6/(x^2+2/3)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T11-6_2.2.1.1_4.1.1', 'plane_constant': '32', 'plane_map_constant_factored': '\\frac{1}{2^{5}} ', 'plane_model': '(-1)*t + x^4*(x^2 + 6)', 'plane_model_latex': '\\left(-1\\right) t + x^{4} \\left(x^{2} + 6\\right)', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[5, 6, 4, 2, 3, 1], [5, 4, 3, 2, 1, 6], [6, 2, 1, 3, 5, 4]]], 'triples_cyc': [['(1,5,3,4,2,6)', '(1,5)(2,4)', '(1,6,4,3)']]}
58. label: 6T11-6_4.1.1_4.2-a
    {'BelyiDB_label': '6T11-[6,4,4]-6-411-42-g1-a', 'BelyiDB_plabel': '6T11-[6,4,4]-6-411-42-g1', 'a_s': 4, 'abc': [6, 4, 4], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^3-175/432*x-625/11664', 'curve_label': '7200.bx3', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/7200/bx/3'], 'g': 1, 'geomtype': 'H', 'group': '6T11', 'group_num': 11, 'is_primitive': False, 'label': '6T11-6_4.1.1_4.2-a', 'lambdas': [[6], [4, 1, 1], [4, 2]], 'map': '125/432/(x^3-5/6*x^2-125/432*x+3125/11664)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T11-6_4.1.1_4.2', 'plane_constant': '15', 'plane_map_constant_factored': '\\frac{1}{3 \\cdot 5} ', 'plane_model': '15*t^2 - 3*(x^4 + 75)*t + 4*x^6', 'plane_model_latex': '15 t^{2} - 3 \\left(x^{4} + 75\\right) t + 4 x^{6}', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[6, 1, 2, 3, 4, 5], [6, 2, 1, 3, 5, 4], [2, 4, 6, 5, 1, 3]]], 'triples_cyc': [['(1,6,5,4,3,2)', '(1,6,4,3)', '(1,2,4,5)(3,6)']]}
59. label: 6T11-6_4.2_2.2.2-a
    {'BelyiDB_label': '6T11-[6,4,2]-6-42-222-g1-a', 'BelyiDB_plabel': '6T11-[6,4,2]-6-42-222-g1', 'a_s': 2, 'abc': [6, 4, 2], 'aut_group': [[4, 5, 6, 1, 2, 3]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^3-52/27*x-560/729', 'curve_label': '576.d4', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/576/d/4'], 'g': 1, 'geomtype': 'H', 'group': '6T11', 'group_num': 11, 'is_primitive': False, 'label': '6T11-6_4.2_2.2.2-a', 'lambdas': [[6], [4, 2], [2, 2, 2]], 'map': '32/27/(x^3-2/3*x^2-32/27*x+640/729)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T11-6_4.2_2.2.2', 'plane_constant': '3', 'plane_map_constant_factored': '\\frac{1}{3} ', 'plane_model': '(-6)*t^2 - 3*(2*x^4 + 9*x^2 + 6)*t + x^6', 'plane_model_latex': '\\left(-6\\right) t^{2} - 3 \\left(2 x^{4} + 9 x^{2} + 6\\right) t + x^{6}', 'primitivization': '3T2-3_2.1_2.1-a', 'triples': [[[3, 4, 2, 6, 1, 5], [2, 4, 6, 5, 1, 3], [4, 6, 5, 1, 3, 2]]], 'triples_cyc': [['(1,3,2,4,6,5)', '(1,2,4,5)(3,6)', '(1,4)(2,6)(3,5)']]}
60. label: 6T12-5.1_2.2.1.1_3.3-a
    {'BelyiDB_label': '6T12-[5,2,3]-51-2211-33-g0-a', 'BelyiDB_plabel': '6T12-[5,2,3]-51-2211-33-g0', 'a_s': 2, 'abc': [5, 2, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'S', 'group': '6T12', 'group_num': 12, 'is_primitive': True, 'label': '6T12-5.1_2.2.1.1_3.3-a', 'lambdas': [[5, 1], [2, 2, 1, 1], [3, 3]], 'map': '-2/9*x/(x^6-5*x^5+35/4*x^4-325/54*x^3+175/144*x^2-125/1296*x+125/46656)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T12-5.1_2.2.1.1_3.3', 'plane_constant': '1728', 'plane_map_constant_factored': '\\frac{1}{2^{6} \\cdot 3^{3}} ', 'plane_model': 'x*t + (x^2 - 22*x + 125)*(x^2 - 4*x - 1)^2', 'plane_model_latex': 'x t + \\left(x^{2} - 22 x + 125\\right) \\left(x^{2} - 4 x - 1\\right)^{2}', 'primitivization': '6T12-5.1_2.2.1.1_3.3-a', 'triples': [[[2, 3, 4, 6, 5, 1], [4, 2, 3, 1, 6, 5], [5, 4, 2, 3, 6, 1]]], 'triples_cyc': [['(1,2,3,4,6)', '(1,4)(5,6)', '(1,5,6)(2,4,3)']]}
61. label: 6T12-5.1_5.1_2.2.1.1-a
    {'BelyiDB_label': '6T12-[5,5,2]-51-51-2211-g0-a', 'BelyiDB_plabel': '6T12-[5,5,2]-51-51-2211-g0', 'a_s': 2, 'abc': [5, 5, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T12', 'group_num': 12, 'is_primitive': True, 'label': '6T12-5.1_5.1_2.2.1.1-a', 'lambdas': [[5, 1], [5, 1], [2, 2, 1, 1]], 'map': '(128/125*x^6+128/125*x^5)/(x^6+4/5*x^5-4/5*x^4-32/25*x^3-16/25*x^2+64/125*x+64/125)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T12-5.1_5.1_2.2.1.1', 'plane_constant': '1/64', 'plane_map_constant_factored': '2^{6}', 'plane_model': '(4*x - 1)*t + x^5*(x + 1)', 'plane_model_latex': '\\left(4 x - 1\\right) t + x^{5} \\left(x + 1\\right)', 'primitivization': '6T12-5.1_5.1_2.2.1.1-a', 'triples': [[[2, 3, 4, 6, 5, 1], [5, 1, 3, 2, 6, 4], [5, 2, 4, 3, 1, 6]]], 'triples_cyc': [['(1,2,3,4,6)', '(1,5,6,4,2)', '(1,5)(3,4)']]}
62. label: 6T12-5.1_3.3_3.3-a
    {'BelyiDB_label': '6T12-[5,3,3]-51-33-33-g1-a', 'BelyiDB_plabel': '6T12-[5,3,3]-51-33-33-g1', 'a_s': 3, 'abc': [5, 3, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^3-216051019069/58773123072*x+152400737745778247/37018604205637632', 'curve_label': '180.a2', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/180/a/2'], 'g': 1, 'geomtype': 'H', 'group': '6T12', 'group_num': 12, 'is_primitive': True, 'label': '6T12-5.1_3.3_3.3-a', 'lambdas': [[5, 1], [3, 3], [3, 3]], 'map': '(418227202051/88159684608*x^2+18619893262512571/1156831381426176*x+174913992535407978606601/15544259980364064227328)/(x^6+311647/69984*x^5+128837763665/58773123072*x^4-182227865341651465/18509302102818816*x^3-33453710528492148375715/10362839986909376151552*x^2+21011893181300954246075158327/2175698980931597341770645504*x-17506662145505095957518012603972011/5481508229334608701193166778269696)*y+(-174913992535407978606601/15544259980364064227328*x-85660804478357884770989314331/3263548471397396012655968256)/(x^6+311647/69984*x^5+128837763665/58773123072*x^4-182227865341651465/18509302102818816*x^3-33453710528492148375715/10362839986909376151552*x^2+21011893181300954246075158327/2175698980931597341770645504*x-17506662145505095957518012603972011/5481508229334608701193166778269696)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T12-5.1_3.3_3.3', 'plane_constant': '-2/81', 'plane_map_constant_factored': '\\frac{3^{4}}{2} ', 'plane_model': '(-225)*t^2 + 2*(-30*x^3 + 8*x - 1)*t + x^5*(x - 2)', 'plane_model_latex': '\\left(-225\\right) t^{2} + 2 \\left(-30 x^{3} + 8 x - 1\\right) t + x^{5} \\left(x - 2\\right)', 'primitivization': '6T12-5.1_3.3_3.3-a', 'triples': [[[4, 1, 5, 3, 2, 6], [3, 1, 2, 6, 4, 5], [3, 6, 5, 2, 1, 4]]], 'triples_cyc': [['(1,4,3,5,2)', '(1,3,2)(4,6,5)', '(1,3,5)(2,6,4)']]}
63. label: 6T12-5.1_5.1_3.3-a
    {'BelyiDB_label': '6T12-[5,5,3]-51-51-33-g1-a', 'BelyiDB_plabel': '6T12-[5,5,3]-51-51-33-g1', 'a_s': 3, 'abc': [5, 5, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^3+7/6000*x+41/540000', 'curve_label': '300.a2', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/300/a/2'], 'g': 1, 'geomtype': 'H', 'group': '6T12', 'group_num': 12, 'is_primitive': True, 'label': '6T12-5.1_5.1_3.3-a', 'lambdas': [[5, 1], [5, 1], [3, 3]], 'map': '(-27/6250*x^2-9/31250*x+2127/12500000)/(x^6-7/10*x^5+901/6000*x^4-1771/270000*x^3-71179/108000000*x^2-43687/3240000000*x-493039/5832000000000)*y+(27/6250*x^3-459/625000*x^2-63/12500000*x+1739/1250000000)/(x^6-7/10*x^5+901/6000*x^4-1771/270000*x^3-71179/108000000*x^2-43687/3240000000*x-493039/5832000000000)', 'orbit_size': 1, 'pass_size': 2, 'plabel': '6T12-5.1_5.1_3.3', 'plane_constant': '27/2', 'plane_map_constant_factored': '\\frac{2}{3^{3}} ', 'plane_model': 'x^5*(x + 2)*t^2 - 2*(-10*x^3 + 2*x - 1)*t - (-18*x - 11)', 'plane_model_latex': 'x^{5} \\left(x + 2\\right) t^{2} - 2 \\left(-10 x^{3} + 2 x - 1\\right) t - \\left(-18 x - 11\\right)', 'primitivization': '6T12-5.1_5.1_3.3-a', 'triples': [[[2, 3, 4, 6, 5, 1], [2, 6, 1, 4, 3, 5], [2, 3, 1, 5, 6, 4]]], 'triples_cyc': [['(1,2,3,4,6)', '(1,2,6,5,3)', '(1,2,3)(4,5,6)']]}
64. label: 6T12-5.1_5.1_3.3-b
    {'BelyiDB_label': '6T12-[5,5,3]-51-51-33-g1-b', 'BelyiDB_plabel': '6T12-[5,5,3]-51-51-33-g1', 'a_s': 3, 'abc': [5, 5, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^3-481/1875*x-9758/421875', 'curve_label': '15.a5', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/15/a/5'], 'g': 1, 'geomtype': 'H', 'group': '6T12', 'group_num': 12, 'is_primitive': True, 'label': '6T12-5.1_5.1_3.3-b', 'lambdas': [[5, 1], [5, 1], [3, 3]], 'map': '(-864/3125*x^2-4032/78125*x-4704/1953125)/(x^6-22/25*x^5-203/375*x^4+37444/84375*x^3+304297/2109375*x^2-49434022/791015625*x-3368254499/177978515625)*y+(864/3125*x^3+6048/78125*x^2-675936/9765625*x-4672928/244140625)/(x^6-22/25*x^5-203/375*x^4+37444/84375*x^3+304297/2109375*x^2-49434022/791015625*x-3368254499/177978515625)', 'orbit_size': 1, 'pass_size': 2, 'plabel': '6T12-5.1_5.1_3.3', 'plane_constant': '1/27', 'plane_map_constant_factored': '3^{3}', 'plane_model': '(-9*x - 1)*t^2 - 5*x^3*t + x^5*(x - 1)', 'plane_model_latex': '\\left(-9 x - 1\\right) t^{2} - 5 x^{3} t + x^{5} \\left(x - 1\\right)', 'primitivization': '6T12-5.1_5.1_3.3-b', 'triples': [[[2, 3, 4, 6, 5, 1], [2, 4, 3, 6, 1, 5], [4, 5, 1, 3, 6, 2]]], 'triples_cyc': [['(1,2,3,4,6)', '(1,2,4,6,5)', '(1,4,3)(2,5,6)']]}
65. label: 6T12-5.1_5.1_5.1-a
    {'BelyiDB_label': '6T12-[5,5,5]-51-51-51-g1-a', 'BelyiDB_plabel': '6T12-[5,5,5]-51-51-51-g1', 'a_s': 5, 'abc': [5, 5, 5], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^3+7780091/80621568*x+22008462877/1880739938304', 'curve_label': '20.a4', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/20/a/4'], 'g': 1, 'geomtype': 'H', 'group': '6T12', 'group_num': 12, 'is_primitive': True, 'label': '6T12-5.1_5.1_5.1-a', 'lambdas': [[5, 1], [5, 1], [5, 1]], 'map': '(-20511149/120932352*x^2+17249876309/235092492288*x+420707233300201/29249267520503808)/(x^6-841/864*x^5+17682025/80621568*x^4-20818816235/940369969152*x^3+2501232064805/2166612408926208*x^2-4627779566302211/151628202826291740672*x+4599592181671097533/14148730862126934905585664)*y+(-420707233300201/29249267520503808*x-353814783205469041/227442304239437611008)/(x^6-841/864*x^5+17682025/80621568*x^4-20818816235/940369969152*x^3+2501232064805/2166612408926208*x^2-4627779566302211/151628202826291740672*x+4599592181671097533/14148730862126934905585664)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T12-5.1_5.1_5.1', 'plane_constant': '2', 'plane_map_constant_factored': '\\frac{1}{2} ', 'plane_model': '(2*x - 1)*t^2 - 2*(2*x - 1)*t + x^5*(x + 2)', 'plane_model_latex': '\\left(2 x - 1\\right) t^{2} - 2 \\left(2 x - 1\\right) t + x^{5} \\left(x + 2\\right)', 'primitivization': '6T12-5.1_5.1_5.1-a', 'triples': [[[2, 3, 4, 6, 5, 1], [6, 5, 3, 1, 4, 2], [1, 4, 6, 3, 2, 5]]], 'triples_cyc': [['(1,2,3,4,6)', '(1,6,2,5,4)', '(2,4,3,6,5)']]}
66. label: 6T13-4.2_3.2.1_2.2.2-a
    {'BelyiDB_label': '6T13-[4,6,2]-42-321-222-g0-a', 'BelyiDB_plabel': '6T13-[4,6,2]-42-321-222-g0', 'a_s': 2, 'abc': [4, 6, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T13', 'group_num': 13, 'is_primitive': False, 'label': '6T13-4.2_3.2.1_2.2.2-a', 'lambdas': [[4, 2], [3, 2, 1], [2, 2, 2]], 'map': '(6561/6241*x^6-15309/6241*x^5+35721/24964*x^4)/(x^6-147/79*x^5-24843/24964*x^4+24353/6241*x^3-7203/6241*x^2-50421/24964*x+117649/99856)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T13-4.2_3.2.1_2.2.2', 'plane_constant': '16', 'plane_map_constant_factored': '\\frac{1}{2^{4}} ', 'plane_model': '(x - 1)*(x + 2)^2*t + x^4*(x + 3)^2', 'plane_model_latex': '\\left(x - 1\\right) \\left(x + 2\\right)^{2} t + x^{4} \\left(x + 3\\right)^{2}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[2, 5, 6, 1, 4, 3], [3, 6, 1, 2, 5, 4], [6, 3, 2, 5, 4, 1]]], 'triples_cyc': [['(1,2,5,4)(3,6)', '(1,3)(2,6,4)', '(1,6)(2,3)(4,5)']]}
67. label: 6T13-6_2.1.1.1.1_4.2-a
    {'BelyiDB_label': '6T13-[6,2,4]-6-21111-42-g0-a', 'BelyiDB_plabel': '6T13-[6,2,4]-6-21111-42-g0', 'a_s': 2, 'abc': [6, 2, 4], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T13', 'group_num': 13, 'is_primitive': False, 'label': '6T13-6_2.1.1.1.1_4.2-a', 'lambdas': [[6], [2, 1, 1, 1, 1], [4, 2]], 'map': '4*x^6/(x^6-6*x^5+9*x^4+8*x^3-24*x^2+16)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T13-6_2.1.1.1.1_4.2', 'plane_constant': '-16/729', 'plane_map_constant_factored': '\\frac{3^{6}}{2^{4}} ', 'plane_model': 't + x^4*(x - 1)^2', 'plane_model_latex': 't + x^{4} \\left(x - 1\\right)^{2}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[6, 1, 4, 5, 2, 3], [3, 2, 1, 4, 5, 6], [2, 5, 6, 1, 4, 3]]], 'triples_cyc': [['(1,6,3,4,5,2)', '(1,3)', '(1,2,5,4)(3,6)']]}
68. label: 6T13-6_4.2_3.2.1-a
    {'BelyiDB_label': '6T13-[6,4,6]-6-42-321-g1-a', 'BelyiDB_plabel': '6T13-[6,4,6]-6-42-321-g1', 'a_s': 4, 'abc': [6, 4, 6], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 6, 'base_field': [-2, 0, 1], 'base_field_label': '2.2.8.1', 'c_s': 6, 'curve': 'y^2=x^3+1/432*(-96*nu-137)*x+1/23328*(3504*nu+4955)', 'curve_label': '2.2.8.1-1458.1-x1', 'deg': 6, 'embeddings': [[-1.414213562373095, 0.0], [1.414213562373095, 0.0]], 'friends': ['EllipticCurve/2.2.8.1/1458.1/x/1'], 'g': 1, 'geomtype': 'H', 'group': '6T13', 'group_num': 13, 'is_primitive': False, 'label': '6T13-6_4.2_3.2.1-a', 'lambdas': [[6], [4, 2], [3, 2, 1]], 'map': '(1/729*(192*nu+272)*x+1/19683*(2192*nu+3100))/(x^6-1/2*x^5+1/432*(-256*nu-339)*x^4+1/11664*(-1280*nu-1799)*x^3+1/186624*(9728*nu+13703)*x^2+1/1119744*(-5632*nu-7963)*x+1/80621568*(12032*nu+17051))*y+(1/729*(-192*nu-272)*x^3+1/729*(-48*nu-68)*x^2+1/19683*(2180*nu+3083)*x+1/2125764*(154732*nu+218825))/(x^6-1/2*x^5+1/432*(-256*nu-339)*x^4+1/11664*(-1280*nu-1799)*x^3+1/186624*(9728*nu+13703)*x^2+1/1119744*(-5632*nu-7963)*x+1/80621568*(12032*nu+17051))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '6T13-6_4.2_3.2.1', 'plane_constant': '1/16*(-nu - 1)', 'plane_map_constant_factored': '2^{4} (-\\nu + 1)', 'plane_model': '8*t^2 - (8*nu*x^3 + (9*nu - 9)*x^2 + (3*nu - 6)*x - 1)*t + x^3*(x - 1)*(2*x + 1)^2', 'plane_model_latex': '8 t^{2} - \\left(8 \\nu x^{3} + \\left(9 \\nu - 9\\right) x^{2} + \\left(3 \\nu - 6\\right) x - 1\\right) t + x^{3} \\left(x - 1\\right) \\left(2 x + 1\\right)^{2}', 'primitivization': '2T1-2_2_1.1-a', 'triples': [[[2, 5, 4, 1, 6, 3], [2, 5, 6, 1, 4, 3], [5, 4, 3, 6, 1, 2]], [[2, 3, 4, 5, 6, 1], [2, 5, 6, 1, 4, 3], [3, 4, 1, 6, 5, 2]]], 'triples_cyc': [['(1,2,5,6,3,4)', '(1,2,5,4)(3,6)', '(1,5)(2,4,6)'], ['(1,2,3,4,5,6)', '(1,2,5,4)(3,6)', '(1,3)(2,4,6)']]}
69. label: 6T14-4.1.1_4.1.1_3.3-a
    {'BelyiDB_label': '6T14-[4,4,3]-411-411-33-g0-a', 'BelyiDB_plabel': '6T14-[4,4,3]-411-411-33-g0', 'a_s': 3, 'abc': [4, 4, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-4.1.1_4.1.1_3.3-a', 'lambdas': [[4, 1, 1], [4, 1, 1], [3, 3]], 'map': '(1140625/117649*x^6+1518750/117649*x^5+506250/117649*x^4)/(x^6-162/49*x^5+810/2401*x^4+699840/117649*x^3-43740/117649*x^2-472392/117649*x-157464/117649)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T14-4.1.1_4.1.1_3.3', 'plane_constant': '27/4', 'plane_map_constant_factored': '\\frac{2^{2}}{3^{3}} ', 'plane_model': '(-1)*(-10*x^2 + 6*x - 1)*t + x^4*(2*x^2 + 18*x + 45)', 'plane_model_latex': '\\left(-1\\right) \\left(-10 x^{2} + 6 x - 1\\right) t + x^{4} \\left(2 x^{2} + 18 x + 45\\right)', 'primitivization': '6T14-4.1.1_4.1.1_3.3-a', 'triples': [[[4, 2, 6, 3, 5, 1], [2, 5, 3, 4, 6, 1], [5, 1, 4, 6, 2, 3]]], 'triples_cyc': [['(1,4,3,6)', '(1,2,5,6)', '(1,5,2)(3,4,6)']]}
70. label: 6T14-5.1_4.1.1_2.2.2-a
    {'BelyiDB_label': '6T14-[5,4,2]-51-411-222-g0-a', 'BelyiDB_plabel': '6T14-[5,4,2]-51-411-222-g0', 'a_s': 2, 'abc': [5, 4, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-5.1_4.1.1_2.2.2-a', 'lambdas': [[5, 1], [4, 1, 1], [2, 2, 2]], 'map': '(512/529*x^6+1792/529*x^5)/(x^6+98/23*x^5+4655/529*x^4+20580/529*x^3+36015/529*x^2+33614/529*x+117649/529)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T14-5.1_4.1.1_2.2.2', 'plane_constant': '4', 'plane_map_constant_factored': '\\frac{1}{2^{2}} ', 'plane_model': '(-1)*(2*x + 1)*t + x^4*(x^2 - 2*x + 5)', 'plane_model_latex': '\\left(-1\\right) \\left(2 x + 1\\right) t + x^{4} \\left(x^{2} - 2 x + 5\\right)', 'primitivization': '6T14-5.1_4.1.1_2.2.2-a', 'triples': [[[2, 3, 4, 6, 5, 1], [1, 6, 3, 2, 4, 5], [2, 1, 4, 3, 6, 5]]], 'triples_cyc': [['(1,2,3,4,6)', '(2,6,5,4)', '(1,2)(3,4)(5,6)']]}
71. label: 6T14-5.1_4.1.1_4.1.1-a
    {'BelyiDB_label': '6T14-[5,4,4]-51-411-411-g0-a', 'BelyiDB_plabel': '6T14-[5,4,4]-51-411-411-g0', 'a_s': 4, 'abc': [5, 4, 4], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-5.1_4.1.1_4.1.1-a', 'lambdas': [[5, 1], [4, 1, 1], [4, 1, 1]], 'map': '2*x^5/(x^6+x^5+75/256*x^4-625/65536*x^2+3125/16777216)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T14-5.1_4.1.1_4.1.1', 'plane_constant': '32', 'plane_map_constant_factored': '\\frac{1}{2^{5}} ', 'plane_model': '(-1)*(x - 1)*t + x^4*(x^2 - 6*x + 10)', 'plane_model_latex': '\\left(-1\\right) \\left(x - 1\\right) t + x^{4} \\left(x^{2} - 6 x + 10\\right)', 'primitivization': '6T14-5.1_4.1.1_4.1.1-a', 'triples': [[[3, 4, 2, 5, 1, 6], [1, 6, 3, 2, 4, 5], [6, 3, 1, 4, 5, 2]]], 'triples_cyc': [['(1,3,2,4,5)', '(2,6,5,4)', '(1,6,2,3)']]}
72. label: 6T14-6_4.1.1_2.2.1.1-a
    {'BelyiDB_label': '6T14-[6,4,2]-6-411-2211-g0-a', 'BelyiDB_plabel': '6T14-[6,4,2]-6-411-2211-g0', 'a_s': 2, 'abc': [6, 4, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-6_4.1.1_2.2.1.1-a', 'lambdas': [[6], [4, 1, 1], [2, 2, 1, 1]], 'map': '2500/2527*x^6/(x^6+162/665*x^5+729/361*x^4+17496/2527*x^3+17496/2527*x^2+314928/12635)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T14-6_4.1.1_2.2.1.1', 'plane_constant': '16/84375', 'plane_map_constant_factored': '\\frac{3^{3} \\cdot 5^{5}}{2^{4}} ', 'plane_model': 'x^6*t - (x^2 - 6*x + 25)', 'plane_model_latex': 'x^{6} t - \\left(x^{2} - 6 x + 25\\right)', 'primitivization': '6T14-6_4.1.1_2.2.1.1-a', 'triples': [[[6, 4, 2, 1, 3, 5], [1, 3, 6, 2, 5, 4], [6, 2, 5, 4, 3, 1]]], 'triples_cyc': [['(1,6,5,3,2,4)', '(2,3,6,4)', '(1,6)(3,5)']]}
73. label: 6T14-6_3.3_4.1.1-a
    {'BelyiDB_label': '6T14-[6,3,4]-6-33-411-g1-a', 'BelyiDB_plabel': '6T14-[6,3,4]-6-33-411-g1', 'a_s': 3, 'abc': [6, 3, 4], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-6, 0, 1], 'base_field_label': '2.2.24.1', 'c_s': 6, 'curve': 'y^2=x^3+1/2430000*(-58944*nu+144479)*x+1/9841500000*(-1321632*nu+3242737)', 'curve_label': '2.2.24.1-10.1-b3', 'deg': 6, 'embeddings': [[-2.449489742783178, 0.0], [2.449489742783178, 0.0]], 'friends': ['EllipticCurve/2.2.24.1/10.1/b/3'], 'g': 1, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-6_3.3_4.1.1-a', 'lambdas': [[6], [3, 3], [4, 1, 1]], 'map': '(1/102515625*(-1480448*nu+3621168)*x+1/115330078125*(-35127104*nu+85853164))/(x^6+1/450*(-64*nu-1)*x^5+1/2430000*(-255872*nu+723077)*x^4+1/196830000*(791616*nu-2171981)*x^3+1/28343520000*(-1599232*nu+4346837)*x^2+1/1530550080000*(535744*nu-1449529)*x+1/991796451840000*(-806016*nu+2174881))*y+(1/102515625*(1480448*nu-3621168)*x^3+1/23066015625*(224118976*nu-548599116)*x^2+1/15569560546875*(16247041328*nu-39797119623)*x+1/126113440429687500*(991602300016*nu-2428740737631))/(x^6+1/450*(-64*nu-1)*x^5+1/2430000*(-255872*nu+723077)*x^4+1/196830000*(791616*nu-2171981)*x^3+1/28343520000*(-1599232*nu+4346837)*x^2+1/1530550080000*(535744*nu-1449529)*x+1/991796451840000*(-806016*nu+2174881))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '6T14-6_3.3_4.1.1', 'plane_constant': '1/625*(282*nu - 738)', 'plane_map_constant_factored': '\\frac{1}{2 \\cdot 3^{2}} (-47 \\nu - 123)', 'plane_model': '8*x^6*t^2 + ((-8*nu - 8)*x^3 + (15*nu + 45)*x^2 + (-18*nu - 42)*x + 7*nu + 13)*t - (2*nu - 5)', 'plane_model_latex': '8 x^{6} t^{2} - \\left(\\left(8 \\nu + 8\\right) x^{3} + \\left(-15 \\nu - 45\\right) x^{2} + \\left(18 \\nu + 42\\right) x - 7 \\nu - 13\\right) t - \\left(2 \\nu - 5\\right)', 'primitivization': '6T14-6_3.3_4.1.1-a', 'triples': [[[2, 3, 6, 1, 4, 5], [5, 1, 4, 6, 2, 3], [3, 2, 5, 1, 4, 6]], [[5, 1, 6, 3, 4, 2], [5, 1, 4, 6, 2, 3], [5, 4, 3, 1, 2, 6]]], 'triples_cyc': [['(1,2,3,6,5,4)', '(1,5,2)(3,4,6)', '(1,3,5,4)'], ['(1,5,4,3,6,2)', '(1,5,2)(3,4,6)', '(1,5,2,4)']]}
74. label: 6T14-6_4.1.1_5.1-a
    {'BelyiDB_label': '6T14-[6,4,5]-6-411-51-g1-a', 'BelyiDB_plabel': '6T14-[6,4,5]-6-411-51-g1', 'a_s': 4, 'abc': [6, 4, 5], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [-6, 0, 1], 'base_field_label': '2.2.24.1', 'c_s': 6, 'curve': 'y^2=x^3+1/314928*(-4928*nu+6527)*x+1/459165024*(736992*nu+6018497)', 'curve_label': '2.2.24.1-750.1-l1', 'deg': 6, 'embeddings': [[2.449489742783178, 0.0], [-2.449489742783178, 0.0]], 'friends': ['EllipticCurve/2.2.24.1/750.1/l/1'], 'g': 1, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-6_4.1.1_5.1-a', 'lambdas': [[6], [4, 1, 1], [5, 1]], 'map': '(50000/531441*x+1/129140163*(2000000*nu-812500))/(x^6+1/162*(64*nu-113)*x^5+1/314928*(-95360*nu+123365)*x^4+1/229582512*(8303040*nu-72734485)*x^3+1/297538935552*(-8256098560*nu+3761396165)*x^2+1/144603922678272*(67379893568*nu-1059343199537)*x+1/843330077059682304*(-109285024839552*nu+42356829141793))*y+(-50000/531441*x^3+1/43046721*(-800000*nu-612500)*x^2+1/10460353203*(16400000*nu-490625)*x+1/30502389939948*(-4800600000*nu+128693134375))/(x^6+1/162*(64*nu-113)*x^5+1/314928*(-95360*nu+123365)*x^4+1/229582512*(8303040*nu-72734485)*x^3+1/297538935552*(-8256098560*nu+3761396165)*x^2+1/144603922678272*(67379893568*nu-1059343199537)*x+1/843330077059682304*(-109285024839552*nu+42356829141793))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '6T14-6_4.1.1_5.1', 'plane_constant': '1/108*(2*nu + 5)', 'plane_map_constant_factored': '2^{2} \\cdot 3^{3} (-2 \\nu + 5)', 'plane_model': '(-216*nu + 540)*x^6*t^2 - (x^6 + (60*nu - 165)*x^2 + (-24*nu + 108)*x - 20)*t + (2*x + 5)', 'plane_model_latex': '\\left(-216 \\nu + 540\\right) x^{6} t^{2} - \\left(x^{6} + \\left(60 \\nu - 165\\right) x^{2} + \\left(-24 \\nu + 108\\right) x - 20\\right) t + \\left(2 x + 5\\right)', 'primitivization': '6T14-6_4.1.1_5.1-a', 'triples': [[[6, 4, 2, 1, 3, 5], [1, 4, 3, 5, 6, 2], [2, 3, 4, 6, 5, 1]], [[4, 1, 2, 5, 6, 3], [5, 2, 3, 6, 4, 1], [2, 3, 4, 6, 5, 1]]], 'triples_cyc': [['(1,6,5,3,2,4)', '(2,4,5,6)', '(1,2,3,4,6)'], ['(1,4,5,6,3,2)', '(1,5,4,6)', '(1,2,3,4,6)']]}
75. label: 6T14-6_5.1_2.2.2-a
    {'BelyiDB_label': '6T14-[6,5,2]-6-51-222-g1-a', 'BelyiDB_plabel': '6T14-[6,5,2]-6-51-222-g1', 'a_s': 2, 'abc': [6, 5, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^3+1679/314928*x-36631/459165024', 'curve_label': '15.a8', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/15/a/8'], 'g': 1, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-6_5.1_2.2.2-a', 'lambdas': [[6], [5, 1], [2, 2, 2]], 'map': '(-3125/2125764*x+15625/2066242608)/(x^6-89/162*x^5+21605/314928*x^4-203845/229582512*x^3+229903205/297538935552*x^2+1372454551/144603922678272*x+1617905136961/843330077059682304)*y+(3125/2125764*x^3+228125/688747536*x^2+3715625/669462604992*x+3972715625/1952152956156672)/(x^6-89/162*x^5+21605/314928*x^4-203845/229582512*x^3+229903205/297538935552*x^2+1372454551/144603922678272*x+1617905136961/843330077059682304)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T14-6_5.1_2.2.2', 'plane_constant': '1/4', 'plane_map_constant_factored': '2^{2}', 'plane_model': 'x^6*t^2 - (-5*x^3 + 12*x + 5)*t + (-3*x + 5)', 'plane_model_latex': 'x^{6} t^{2} - \\left(-5 x^{3} + 12 x + 5\\right) t + \\left(-3 x + 5\\right)', 'primitivization': '6T14-6_5.1_2.2.2-a', 'triples': [[[6, 4, 2, 1, 3, 5], [3, 4, 2, 5, 1, 6], [2, 1, 4, 3, 6, 5]]], 'triples_cyc': [['(1,6,5,3,2,4)', '(1,3,2,4,5)', '(1,2)(3,4)(5,6)']]}
76. label: 6T14-6_6_2.2.1.1-a
    {'BelyiDB_label': '6T14-[6,6,2]-6-6-2211-g1-a', 'BelyiDB_plabel': '6T14-[6,6,2]-6-6-2211-g1', 'a_s': 2, 'abc': [6, 6, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^3-21769/34992*x+437147/17006112', 'curve_label': '90.c2', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/90/c/2'], 'g': 1, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-6_6_2.2.1.1-a', 'lambdas': [[6], [6], [2, 2, 1, 1]], 'map': '(12500/19683*x-109375/1594323)/(x^6+37/54*x^5+6845/34992*x^4-10546735/8503056*x^3-3921829195/3673320192*x^2+276095743957/595077871104*x+441020695326409/1156831381426176)*y+(-12500/19683*x^3-284375/531441*x^2+39934375/172186884*x+65652353125/167365651248)/(x^6+37/54*x^5+6845/34992*x^4-10546735/8503056*x^3-3921829195/3673320192*x^2+276095743957/595077871104*x+441020695326409/1156831381426176)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T14-6_6_2.2.1.1', 'plane_constant': '-3125/27', 'plane_map_constant_factored': '\\frac{3^{3}}{5^{5}} ', 'plane_model': 't^2 + (4*x^3 - 9*x^2 - 42*x - 25)*t + 4*x^6', 'plane_model_latex': 't^{2} + \\left(4 x^{3} - 9 x^{2} - 42 x - 25\\right) t + 4 x^{6}', 'primitivization': '6T14-6_6_2.2.1.1-a', 'triples': [[[6, 4, 2, 1, 3, 5], [5, 6, 4, 2, 3, 1], [3, 5, 1, 4, 2, 6]]], 'triples_cyc': [['(1,6,5,3,2,4)', '(1,5,3,4,2,6)', '(1,3)(2,5)']]}
77. label: 6T14-6_6_3.3-a
    {'BelyiDB_label': '6T14-[6,6,3]-6-6-33-g2-a', 'BelyiDB_plabel': '6T14-[6,6,3]-6-6-33-g2', 'a_s': 3, 'abc': [6, 6, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^6+x^5-13/48*x^4+91/768*x^3-43/3072*x^2+25/8192*x-319/2359296', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 2, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-6_6_3.3-a', 'lambdas': [[6], [6], [3, 3]], 'map': '(5/64*x^3+5/64*x^2+35/6144*x-35/98304)/(x^6-3/16*x^5+35/256*x^4-65/4096*x^3+35/6144*x^2-1/3072*x+1/13824)*y+(5/64*x^6+15/128*x^5+25/1024*x^4-25/8192*x^3+125/49152*x^2-5/32768*x+31945/452984832)/(x^6-3/16*x^5+35/256*x^4-65/4096*x^3+35/6144*x^2-1/3072*x+1/13824)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T14-6_6_3.3', 'plane_constant': '4/3', 'plane_map_constant_factored': '\\frac{3}{2^{2}} ', 'plane_model': 'x^6*t^3 + 3*(-5*x^4 + 72*x - 60)*t^2 - 3*(-25*x^2 - 96*x + 80)*t - 125', 'plane_model_latex': 'x^{6} t^{3} + 3 \\left(-5 x^{4} + 72 x - 60\\right) t^{2} - 3 \\left(-25 x^{2} - 96 x + 80\\right) t - 125', 'primitivization': '6T14-6_6_3.3-a', 'triples': [[[6, 4, 2, 1, 3, 5], [3, 5, 4, 6, 1, 2], [3, 1, 2, 6, 4, 5]]], 'triples_cyc': [['(1,6,5,3,2,4)', '(1,3,4,6,2,5)', '(1,3,2)(4,6,5)']]}
78. label: 6T14-6_6_5.1-a
    {'BelyiDB_label': '6T14-[6,6,5]-6-6-51-g2-a', 'BelyiDB_plabel': '6T14-[6,6,5]-6-6-51-g2', 'a_s': 5, 'abc': [6, 6, 5], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 6, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'y^2=x^6+x^5+29/16*x^4+93/64*x^3+969/1024*x^2+225/1024*x+153/2048', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 2, 'geomtype': 'H', 'group': '6T14', 'group_num': 14, 'is_primitive': True, 'label': '6T14-6_6_5.1-a', 'lambdas': [[6], [6], [5, 1]], 'map': '(5/2*x^2+5/8*x+65/64)/(x^5+15/4*x^4+45/8*x^3+135/32*x^2+405/256*x+243/1024)*y+(5/2*x^5+15/8*x^4+105/32*x^3+235/128*x^2+2055/2048*x-165/4096)/(x^5+15/4*x^4+45/8*x^3+135/32*x^2+405/256*x+243/1024)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T14-6_6_5.1', 'plane_constant': '4', 'plane_map_constant_factored': '\\frac{1}{2^{2}} ', 'plane_model': '(-1)*t^2 + (-3*x^5 + 15*x^4 - 40*x^3 + 60*x^2 - 48*x + 20)*t + x^6', 'plane_model_latex': '\\left(-1\\right) t^{2} + \\left(-3 x^{5} + 15 x^{4} - 40 x^{3} + 60 x^{2} - 48 x + 20\\right) t + x^{6}', 'primitivization': '6T14-6_6_5.1-a', 'triples': [[[6, 4, 2, 1, 3, 5], [2, 3, 6, 1, 4, 5], [5, 2, 6, 1, 3, 4]]], 'triples_cyc': [['(1,6,5,3,2,4)', '(1,2,3,6,5,4)', '(1,5,3,6,4)']]}
79. label: 6T15-4.2_3.1.1.1_3.3-a
    {'BelyiDB_label': '6T15-[4,3,3]-42-3111-33-g0-a', 'BelyiDB_plabel': '6T15-[4,3,3]-42-3111-33-g0', 'a_s': 3, 'abc': [4, 3, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 4, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-4.2_3.1.1.1_3.3-a', 'lambdas': [[4, 2], [3, 1, 1, 1], [3, 3]], 'map': '(6561/686*x^6+4374/343*x^5+1458/343*x^4)/(x^6-24/7*x^5+24/49*x^4+2176/343*x^3-192/343*x^2-1536/343*x-512/343)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T15-4.2_3.1.1.1_3.3', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '(-1)*(4*x^3 + 9*x^2 - 6*x + 1)*t - x^4*(x + 3)^2', 'plane_model_latex': '\\left(-1\\right) \\left(4 x^{3} + 9 x^{2} - 6 x + 1\\right) t - x^{4} \\left(x + 3\\right)^{2}', 'primitivization': '6T15-4.2_3.1.1.1_3.3-a', 'triples': [[[3, 1, 4, 2, 6, 5], [1, 2, 4, 5, 3, 6], [2, 3, 1, 5, 6, 4]]], 'triples_cyc': [['(1,3,4,2)(5,6)', '(3,4,5)', '(1,2,3)(4,5,6)']]}
80. label: 6T15-5.1_3.3_3.1.1.1-a
    {'BelyiDB_label': '6T15-[5,3,3]-51-33-3111-g0-a', 'BelyiDB_plabel': '6T15-[5,3,3]-51-33-3111-g0', 'a_s': 3, 'abc': [5, 3, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 3, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_3.3_3.1.1.1-a', 'lambdas': [[5, 1], [3, 3], [3, 1, 1, 1]], 'map': '(2187/856*x^6+729/107*x^5)/(x^6+1092/107*x^5+2640/107*x^4-4160/107*x^3-15360/107*x^2+12288/107*x+32768/107)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T15-5.1_3.3_3.1.1.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '(-3*x - 1)*t + x^3*(x^3 - 3*x^2 + 5)', 'plane_model_latex': '\\left(-3 x - 1\\right) t + x^{3} \\left(x^{3} - 3 x^{2} + 5\\right)', 'primitivization': '6T15-5.1_3.3_3.1.1.1-a', 'triples': [[[6, 4, 3, 5, 1, 2], [3, 5, 6, 2, 4, 1], [2, 3, 1, 4, 5, 6]]], 'triples_cyc': [['(1,6,2,4,5)', '(1,3,6)(2,5,4)', '(1,2,3)']]}
81. label: 6T15-5.1_4.2_2.2.1.1-a
    {'BelyiDB_label': '6T15-[5,4,2]-51-42-2211-g0-a', 'BelyiDB_plabel': '6T15-[5,4,2]-51-42-2211-g0', 'a_s': 2, 'abc': [5, 4, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [4, -1, 1], 'base_field_label': '2.0.15.1', 'c_s': 5, 'curve': 'PP1', 'deg': 6, 'embeddings': [[0.5, 1.936491673103708], [0.5, -1.936491673103708]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_4.2_2.2.1.1-a', 'lambdas': [[5, 1], [4, 2], [2, 2, 1, 1]], 'map': '(1/648626449*(67262400*nu+1902382848)*x^6+1/648626449*(-350695008*nu-589286016)*x^5)/(x^6+1/3074059*(2550981*nu+1416912)*x^5+1/2594505796*(-1006504575*nu+57591900)*x^4+1/648626449*(-142471950*nu+949516600)*x^3+1/648626449*(-75218250*nu-327741000)*x^2+1/648626449*(-49794600*nu+31384800)*x+1/648626449*(19666500*nu-43658000))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '6T15-5.1_4.2_2.2.1.1', 'plane_constant': '1/256*(22275*nu - 56700)', 'plane_map_constant_factored': '\\frac{2^{2}}{3^{5} \\cdot 5^{3}} (-11 \\nu - 17)', 'plane_model': '(-1)*(x - 1)^2*t + x^5*(x - 3*nu + 6)', 'plane_model_latex': '\\left(-1\\right) \\left(x - 1\\right)^{2} t + x^{5} \\left(x - 3 \\nu + 6\\right)', 'primitivization': '6T15-5.1_4.2_2.2.1.1-a', 'triples': [[[6, 3, 5, 4, 1, 2], [6, 5, 4, 2, 3, 1], [2, 1, 4, 3, 5, 6]], [[3, 2, 5, 6, 4, 1], [2, 6, 5, 1, 3, 4], [2, 1, 4, 3, 5, 6]]], 'triples_cyc': [['(1,6,2,3,5)', '(1,6)(2,5,3,4)', '(1,2)(3,4)'], ['(1,3,5,4,6)', '(1,2,6,4)(3,5)', '(1,2)(3,4)']]}
82. label: 6T15-5.1_4.2_3.1.1.1-a
    {'BelyiDB_label': '6T15-[5,4,3]-51-42-3111-g0-a', 'BelyiDB_plabel': '6T15-[5,4,3]-51-42-3111-g0', 'a_s': 3, 'abc': [5, 4, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-10, 0, 1], 'base_field_label': '2.2.40.1', 'c_s': 5, 'curve': 'PP1', 'deg': 6, 'embeddings': [[3.162277660168379, 0.0], [-3.162277660168379, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_4.2_3.1.1.1-a', 'lambdas': [[5, 1], [4, 2], [3, 1, 1, 1]], 'map': '(1/141282479*(290611200*nu-775598592)*x^6+1/141282479*(-4183054848*nu+13114983168)*x^5)/(x^6+1/835991*(-4907476*nu+14725606)*x^5+1/10867883*(-700401100*nu+2209555285)*x^4+1/141282479*(-22559562760*nu+71283240980)*x^3+1/141282479*(109642049160*nu-346621073295)*x^2+1/141282479*(20046381180*nu-63290153514)*x+1/141282479*(-311389137836*nu+984618481231))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '6T15-5.1_4.2_3.1.1.1', 'plane_constant': '1/108*(-1475*nu + 9125)', 'plane_map_constant_factored': '\\frac{2^{2}}{3^{6} \\cdot 5^{3}} (59 \\nu + 365)', 'plane_model': '(-1)*x^4*(x + 1)^2*t + ((-2*nu + 2)*x - nu - 1)', 'plane_model_latex': '\\left(-1\\right) x^{4} \\left(x + 1\\right)^{2} t + \\left(\\left(-2 \\nu + 2\\right) x - \\nu - 1\\right)', 'primitivization': '6T15-5.1_4.2_3.1.1.1-a', 'triples': [[[2, 4, 3, 5, 6, 1], [3, 6, 1, 2, 4, 5], [2, 3, 1, 4, 5, 6]], [[4, 2, 5, 3, 6, 1], [4, 6, 2, 1, 3, 5], [2, 3, 1, 4, 5, 6]]], 'triples_cyc': [['(1,2,4,5,6)', '(1,3)(2,6,5,4)', '(1,2,3)'], ['(1,4,3,5,6)', '(1,4)(2,6,5,3)', '(1,2,3)']]}
83. label: 6T15-5.1_5.1_2.2.1.1-a
    {'BelyiDB_label': '6T15-[5,5,2]-51-51-2211-g0-a', 'BelyiDB_plabel': '6T15-[5,5,2]-51-51-2211-g0', 'a_s': 2, 'abc': [5, 5, 2], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [4, -1, 1], 'base_field_label': '2.0.15.1', 'c_s': 5, 'curve': 'PP1', 'deg': 6, 'embeddings': [[0.5, -1.936491673103708], [0.5, 1.936491673103708]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_5.1_2.2.1.1-a', 'lambdas': [[5, 1], [5, 1], [2, 2, 1, 1]], 'map': '(1/355560181*(-1263600*nu+359783856)*x^6+1/355560181*(478448208*nu-113188752)*x^5)/(x^6+1/1547596*(2036253*nu-903657)*x^5+1/1422240724*(-1625955225*nu-1131108075)*x^4+1/711120362*(-3868694625*nu+4217024525)*x^3+1/711120362*(-1818380715*nu+13747790895)*x^2+1/1422240724*(9767319495*nu-23163438267)*x+1/1422240724*(-21117264861*nu-44371224551))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '6T15-5.1_5.1_2.2.1.1', 'plane_constant': '1/497664*(-5*nu + 1)', 'plane_map_constant_factored': '2^{6} \\cdot 3^{4} (5 \\nu - 4)', 'plane_model': 'x^5*(x + 6*nu + 6)*t - (x + nu - 4)', 'plane_model_latex': 'x^{5} \\left(x + 6 \\nu + 6\\right) t - \\left(x + \\nu - 4\\right)', 'primitivization': '6T15-5.1_5.1_2.2.1.1-a', 'triples': [[[2, 3, 5, 4, 6, 1], [1, 6, 4, 2, 3, 5], [2, 1, 4, 3, 5, 6]], [[3, 2, 4, 5, 6, 1], [2, 6, 3, 1, 4, 5], [2, 1, 4, 3, 5, 6]]], 'triples_cyc': [['(1,2,3,5,6)', '(2,6,5,3,4)', '(1,2)(3,4)'], ['(1,3,4,5,6)', '(1,2,6,5,4)', '(1,2)(3,4)']]}
84. label: 6T15-5.1_5.1_3.1.1.1-a
    {'BelyiDB_label': '6T15-[5,5,3]-51-51-3111-g0-a', 'BelyiDB_plabel': '6T15-[5,5,3]-51-51-3111-g0', 'a_s': 3, 'abc': [5, 5, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_5.1_3.1.1.1-a', 'lambdas': [[5, 1], [5, 1], [3, 1, 1, 1]], 'map': '(4374/4375*x^6-729/350*x^5)/(x^6-465/224*x^5+75/896*x^4+125/256*x^3+9375/7168*x^2+46875/57344*x-390625/229376)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T15-5.1_5.1_3.1.1.1', 'plane_constant': '1/11664', 'plane_map_constant_factored': '2^{4} \\cdot 3^{6}', 'plane_model': '(-9*x + 1)*t + x^5*(4*x - 1)', 'plane_model_latex': '\\left(-9 x + 1\\right) t + x^{5} \\left(4 x - 1\\right)', 'primitivization': '6T15-5.1_5.1_3.1.1.1-a', 'triples': [[[3, 2, 4, 5, 6, 1], [1, 6, 2, 3, 4, 5], [2, 3, 1, 4, 5, 6]]], 'triples_cyc': [['(1,3,4,5,6)', '(2,6,5,4,3)', '(1,2,3)']]}
85. label: 6T15-4.2_4.2_4.2-a
    {'BelyiDB_label': '6T15-[4,4,4]-42-42-42-g1-a', 'BelyiDB_plabel': '6T15-[4,4,4]-42-42-42-g1', 'a_s': 4, 'abc': [4, 4, 4], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [1, -1, 1], 'base_field_label': '2.0.3.1', 'c_s': 4, 'curve': 'y^2=x^3+1/81051701175042873165624794787*(4168654751420591981720371200*nu-47087995699792025203794057728)', 'curve_label': '2.0.3.1-36864.1-CMa1', 'deg': 6, 'embeddings': [[0.5, -0.8660254037844387], [0.5, 0.8660254037844387]], 'friends': ['EllipticCurve/2.0.3.1/36864.1/CMa/1'], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-4.2_4.2_4.2-a', 'lambdas': [[4, 2], [4, 2], [4, 2]], 'map': '(1/237111834206702395541703*(595575750159238436962304*nu-1067616722214818755220608)*x^2+1/1026141554109767788567865110268349*(-5561567919348895013693166322515968*nu-7715454938141207876729463903431168)*x+1/4440801078502413646894430685022719800753967*(-43774516606051393356566851801460568765497344*nu+17313354156828310565234667800501092342863872))/(x^6+1/1442556361*(7009357824*nu+219393200)*x^5+1/6242906563984686963*(64086980890176873120*nu-41776189164289165504)*x^4+1/4265879009212782798190778673*(37942643532992234210154324480*nu-42557794693412820344441289472)*x^3+1/116921647099929271132788254259306490107*(460183559122089197094807880935430736640*nu-682550300220780805558954930613678535424)*x^2+1/505998197287800518168092115543543762257309261881*(448020058378254686945363026214019125163310571520*nu-785614306199673937639963263961525664639854342144)*x+1/6569378263368446266642296136601574240535378750235862375369*(529734532979166827623245665356993751054397284629235023872*nu-1029599044872093006086809220970958970829388205131243323392))*y+(1/2080968854661562321*(2872451216659546336*nu+3458153265805365136)*x^4+1/27017233725014291055208264929*(225018341620408646064669360640*nu-96876715492377103100883288320)*x^3+1/38973882366643090377596084753102163369*(262741243999158410544482952654916000000*nu-213097357782493695974856296911963447808)*x^2+1/168666065762600172722697371847847920752436420627*(1664908791184012972951430113254253085722982260736*nu-1559458990589084239088271109615064305149119186944)*x+1/2189792754456148755547432045533858080178459583411954125123*(2763960434604248837113551041113264548028074754760015233024*nu-15513354855172511171440856178462645449593836769935163277312))/(x^6+1/1442556361*(7009357824*nu+219393200)*x^5+1/6242906563984686963*(64086980890176873120*nu-41776189164289165504)*x^4+1/4265879009212782798190778673*(37942643532992234210154324480*nu-42557794693412820344441289472)*x^3+1/116921647099929271132788254259306490107*(460183559122089197094807880935430736640*nu-682550300220780805558954930613678535424)*x^2+1/505998197287800518168092115543543762257309261881*(448020058378254686945363026214019125163310571520*nu-785614306199673937639963263961525664639854342144)*x+1/6569378263368446266642296136601574240535378750235862375369*(529734532979166827623245665356993751054397284629235023872*nu-1029599044872093006086809220970958970829388205131243323392))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '6T15-4.2_4.2_4.2', 'plane_constant': '-6*nu + 3', 'plane_map_constant_factored': '\\frac{1}{3^{2}} (2 \\nu - 1)', 'plane_model': '(-1)*x^4*(x - 187*nu + 188)^2*t^2 - (125*nu + 11656)*(18*x^4 + (-3488*nu + 4008)*x^3 + (-327743*nu + 201652)*x^2 + (-13217906*nu + 4570030)*x - 138776336*nu + 2929625)*t - (102989502000*nu + 3166126120582)', 'plane_model_latex': '\\left(-1\\right) x^{4} \\left(x - 187 \\nu + 188\\right)^{2} t^{2} - \\left(125 \\nu + 11656\\right) \\left(18 x^{4} + \\left(-3488 \\nu + 4008\\right) x^{3} + \\left(-327743 \\nu + 201652\\right) x^{2} + \\left(-13217906 \\nu + 4570030\\right) x - 138776336 \\nu + 2929625\\right) t - \\left(102989502000 \\nu + 3166126120582\\right)', 'primitivization': '6T15-4.2_4.2_4.2-a', 'triples': [[[2, 3, 4, 1, 6, 5], [6, 3, 1, 5, 4, 2], [5, 3, 6, 2, 1, 4]], [[2, 3, 4, 1, 6, 5], [6, 3, 5, 2, 4, 1], [5, 6, 4, 2, 1, 3]]], 'triples_cyc': [['(1,2,3,4)(5,6)', '(1,6,2,3)(4,5)', '(1,5)(2,3,6,4)'], ['(1,2,3,4)(5,6)', '(1,6)(2,3,5,4)', '(1,5)(2,6,3,4)']]}
86. label: 6T15-5.1_4.2_3.3-a
    {'BelyiDB_label': '6T15-[5,4,3]-51-42-33-g1-a', 'BelyiDB_plabel': '6T15-[5,4,3]-51-42-33-g1', 'a_s': 3, 'abc': [5, 4, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-10, 0, 1], 'base_field_label': '2.2.40.1', 'c_s': 5, 'curve': 'y^2=x^3+1/46875*(-15884032*nu-50229689)*x+1/52734375*(515007865088*nu+1628597868634)', 'deg': 6, 'embeddings': [[-3.162277660168379, 0.0], [3.162277660168379, 0.0]], 'friends': [], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_4.2_3.3-a', 'lambdas': [[5, 1], [4, 2], [3, 3]], 'map': '(1/390625*(890044416*nu+2814566400)*x^2+1/48828125*(4429534593024*nu+14007418306560)*x+1/6103515625*(6774503737196544*nu+21422861828382720))/(x^6+1/125*(1216*nu+3734)*x^5+1/9375*(-1047488*nu-3310087)*x^4+1/10546875*(-116461946752*nu-368285209652)*x^3+1/1318359375*(94825569207424*nu+299864777724977)*x^2+1/2471923828125*(9827955791159757760*nu+31078725043468227254)*x+1/2780914306640625*(-117425818468119327543104*nu-371333042468721457725271))*y+(1/390625*(-890044416*nu-2814566400)*x^3+1/48828125*(-1761334198272*nu-5569827840000)*x^2+1/30517578125*(-424143453578526720*nu-1341259367958061056)*x+1/3814697265625*(-1212572121351820476416*nu-3834489730693837348864))/(x^6+1/125*(1216*nu+3734)*x^5+1/9375*(-1047488*nu-3310087)*x^4+1/10546875*(-116461946752*nu-368285209652)*x^3+1/1318359375*(94825569207424*nu+299864777724977)*x^2+1/2471923828125*(9827955791159757760*nu+31078725043468227254)*x+1/2780914306640625*(-117425818468119327543104*nu-371333042468721457725271))', 'orbit_size': 2, 'pass_size': 2, 'plabel': '6T15-5.1_4.2_3.3', 'plane_constant': '1/1728*(2650*nu + 8375)', 'plane_map_constant_factored': '\\frac{2^{6}}{5^{3}} (106 \\nu - 335)', 'plane_model': '16*t^2 + (nu - 3)*((-8*nu - 64)*x^3 + (-25*nu - 65)*x^2 + (61*nu + 191)*x - 19*nu - 60)*t + x^5*(x + nu + 5)', 'plane_model_latex': '16 t^{2} - \\left(\\nu - 3\\right) \\left(\\left(8 \\nu + 64\\right) x^{3} + \\left(25 \\nu + 65\\right) x^{2} + \\left(-61 \\nu - 191\\right) x + 19 \\nu + 60\\right) t + x^{5} \\left(x + \\nu + 5\\right)', 'primitivization': '6T15-5.1_4.2_3.3-a', 'triples': [[[2, 3, 4, 5, 1, 6], [2, 5, 1, 6, 3, 4], [2, 3, 1, 5, 6, 4]], [[6, 3, 5, 4, 1, 2], [2, 5, 6, 1, 4, 3], [2, 3, 1, 5, 6, 4]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,2,5,3)(4,6)', '(1,2,3)(4,5,6)'], ['(1,6,2,3,5)', '(1,2,5,4)(3,6)', '(1,2,3)(4,5,6)']]}
87. label: 6T15-5.1_4.2_4.2-a
    {'BelyiDB_label': '6T15-[5,4,4]-51-42-42-g1-a', 'BelyiDB_plabel': '6T15-[5,4,4]-51-42-42-g1', 'a_s': 4, 'abc': [5, 4, 4], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [4, -1, 1], 'base_field_label': '2.0.15.1', 'c_s': 5, 'curve': 'y^2=x^3+1/25389989167104*(-2817860383365*nu+50113607537)*x+1/575713332606076452864*(13203026567584805105*nu-22006329533036288221)', 'deg': 6, 'embeddings': [[0.5, -1.936491673103708], [0.5, 1.936491673103708]], 'friends': [], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_4.2_4.2-a', 'lambdas': [[5, 1], [4, 2], [4, 2]], 'map': '(1/38084983750656*(-2399348803035*nu+2196769688287)*x^2+1/287856666303038226432*(-1518734625547514795*nu-4571521267114961393)*x+1/2900931974575463122360860672*(-4784758765644479793890865*nu-18201701651125385755850531))/(x^6+1/2519424*(-94175*nu+594067)*x^5+1/25389989167104*(-4020889389255*nu+1829310594395)*x^4+1/287856666303038226432*(-1165080330468836185*nu-12019669087140662875)*x^3+1/1933954649716975414907240448*(6347449574122852011515355*nu-51036728205009353733882175)*x^2+1/14617355278225623203181778075385856*(33428875726736906893059132950185*nu+39613546408060201998536904691723)*x+1/441927788453859750156636576549611218403328*(-84905461885839945679860314970810823875*nu+85554030047459081587455910876666993927))*y+(1/2900931974575463122360860672*(4784758765644479793890865*nu+18201701651125385755850531)*x+1/21926032917338434804772667113078784*(-24585828120206211144746043437795*nu+11390604780004818210868287094951))/(x^6+1/2519424*(-94175*nu+594067)*x^5+1/25389989167104*(-4020889389255*nu+1829310594395)*x^4+1/287856666303038226432*(-1165080330468836185*nu-12019669087140662875)*x^3+1/1933954649716975414907240448*(6347449574122852011515355*nu-51036728205009353733882175)*x^2+1/14617355278225623203181778075385856*(33428875726736906893059132950185*nu+39613546408060201998536904691723)*x+1/441927788453859750156636576549611218403328*(-84905461885839945679860314970810823875*nu+85554030047459081587455910876666993927))', 'orbit_size': 2, 'pass_size': 4, 'plabel': '6T15-5.1_4.2_4.2', 'plane_constant': '1/9*(-4*nu + 20)', 'plane_map_constant_factored': '\\frac{3}{2^{5}} (\\nu + 4)', 'plane_model': '(1/2*(3*nu - 6))*(6*x - nu + 3)*t^2 - (4*nu - 4)*(6*x - nu + 3)*t - x^4*(x + 2*nu - 1)^2', 'plane_model_latex': '\\left(1/2 \\left(3 \\nu - 6\\right)\\right) \\left(6 x - \\nu + 3\\right) t^{2} - \\left(4 \\nu - 4\\right) \\left(6 x - \\nu + 3\\right) t - x^{4} \\left(x + 2 \\nu - 1\\right)^{2}', 'primitivization': '6T15-5.1_4.2_4.2-a', 'triples': [[[5, 2, 6, 1, 3, 4], [2, 3, 4, 1, 6, 5], [3, 1, 6, 5, 4, 2]], [[5, 6, 3, 1, 2, 4], [2, 3, 4, 1, 6, 5], [3, 6, 2, 5, 4, 1]]], 'triples_cyc': [['(1,5,3,6,4)', '(1,2,3,4)(5,6)', '(1,3,6,2)(4,5)'], ['(1,5,2,6,4)', '(1,2,3,4)(5,6)', '(1,3,2,6)(4,5)']]}
88. label: 6T15-5.1_4.2_4.2-b
    {'BelyiDB_label': '6T15-[5,4,4]-51-42-42-g1-b', 'BelyiDB_plabel': '6T15-[5,4,4]-51-42-42-g1', 'a_s': 4, 'abc': [5, 4, 4], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-15, 0, 1], 'base_field_label': '2.2.60.1', 'c_s': 5, 'curve': 'y^2=x^3-3483/15625*x-20682/1953125', 'curve_label': '2.2.60.1-90.1-a3', 'deg': 6, 'embeddings': [[-3.872983346207417, 0.0], [3.872983346207417, 0.0]], 'friends': ['EllipticCurve/2.2.60.1/90.1/a/3'], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_4.2_4.2-b', 'lambdas': [[5, 1], [4, 2], [4, 2]], 'map': '(-20736/390625*nu*x^2+2363904/48828125*nu*x+11010816/6103515625*nu)/(x^6-234/125*x^5+1971/3125*x^4+1/390625*(20736*nu+167076)*x^3+1/48828125*(-2426112*nu-8316189)*x^2+1/30517578125*(-198754560*nu-741499434)*x+1/3814697265625*(20051815680*nu+77714730801))*y+(20736/390625*nu*x^3-2426112/48828125*nu*x^2+1/30517578125*(-198754560*nu-644972544)*x+1/3814697265625*(20051815680*nu+76751732736))/(x^6-234/125*x^5+1971/3125*x^4+1/390625*(20736*nu+167076)*x^3+1/48828125*(-2426112*nu-8316189)*x^2+1/30517578125*(-198754560*nu-741499434)*x+1/3814697265625*(20051815680*nu+77714730801))', 'orbit_size': 2, 'pass_size': 4, 'plabel': '6T15-5.1_4.2_4.2', 'plane_constant': '1/36*(-100*nu - 375)', 'plane_map_constant_factored': '\\frac{2^{2} \\cdot 3}{5^{3}} (-4 \\nu + 15)', 'plane_model': '(4*nu + 16)*t^2 + (nu - 4)*(9*x^4 + (-12*nu - 56)*x^3 + (-40*nu - 150)*x^2 + (-40*nu - 156)*x - 692*nu - 2680)*t - x^5*(2*x - nu - 15)', 'plane_model_latex': '\\left(4 \\nu + 16\\right) t^{2} + \\left(\\nu - 4\\right) \\left(9 x^{4} + \\left(-12 \\nu - 56\\right) x^{3} + \\left(-40 \\nu - 150\\right) x^{2} + \\left(-40 \\nu - 156\\right) x - 692 \\nu - 2680\\right) t - x^{5} \\left(2 x - \\nu - 15\\right)', 'primitivization': '6T15-5.1_4.2_4.2-b', 'triples': [[[5, 6, 3, 2, 4, 1], [2, 3, 4, 1, 6, 5], [5, 3, 2, 6, 4, 1]], [[2, 3, 6, 1, 5, 4], [2, 3, 4, 1, 6, 5], [3, 4, 1, 5, 6, 2]]], 'triples_cyc': [['(1,5,4,2,6)', '(1,2,3,4)(5,6)', '(1,5,4,6)(2,3)'], ['(1,2,3,6,4)', '(1,2,3,4)(5,6)', '(1,3)(2,4,5,6)']]}
89. label: 6T15-5.1_5.1_3.3-a
    {'BelyiDB_label': '6T15-[5,5,3]-51-51-33-g1-a', 'BelyiDB_plabel': '6T15-[5,5,3]-51-51-33-g1', 'a_s': 3, 'abc': [5, 5, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 5, 'curve': 'y^2=x^3-243/2000*x-729/100000', 'curve_label': '2700.d2', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': ['EllipticCurve/Q/2700/d/2'], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_5.1_3.3-a', 'lambdas': [[5, 1], [5, 1], [3, 3]], 'map': '(-729/6250*x^2-6561/156250*x-19683/12500000)/(x^6-27/50*x^5-729/2000*x^4+8019/50000*x^3+1121931/20000000*x^2-63950067/5000000000*x-3645153819/1000000000000)*y+(729/6250*x^3+19683/625000*x^2-4074381/312500000*x-112134051/31250000000)/(x^6-27/50*x^5-729/2000*x^4+8019/50000*x^3+1121931/20000000*x^2-63950067/5000000000*x-3645153819/1000000000000)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T15-5.1_5.1_3.3', 'plane_constant': '2', 'plane_map_constant_factored': '\\frac{1}{2} ', 'plane_model': 't^2 - 2*t + x^5*(-5*x - 6)', 'plane_model_latex': 't^{2} - 2 t + x^{5} \\left(-5 x - 6\\right)', 'primitivization': '6T15-5.1_5.1_3.3-a', 'triples': [[[6, 4, 2, 1, 5, 3], [6, 4, 3, 1, 2, 5], [2, 3, 1, 5, 6, 4]]], 'triples_cyc': [['(1,6,3,2,4)', '(1,6,5,2,4)', '(1,2,3)(4,5,6)']]}
90. label: 6T15-5.1_5.1_4.2-a
    {'BelyiDB_label': '6T15-[5,5,4]-51-51-42-g1-a', 'BelyiDB_plabel': '6T15-[5,5,4]-51-51-42-g1', 'a_s': 4, 'abc': [5, 5, 4], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [16, 0, -2, 0, 1], 'base_field_label': '4.0.14400.3', 'c_s': 5, 'curve': 'y^2=x^3+1/295245000*(187132*nu^3-52599*nu^2-1245024*nu-2424448)*x+1/16142520375000*(3341164270*nu^3-5720990089*nu^2-38301326128*nu-65046518024)', 'deg': 6, 'embeddings': [[1.58113883008419, 1.224744871391589], [-1.58113883008419, 1.224744871391589], [-1.58113883008419, -1.224744871391589], [1.58113883008419, -1.224744871391589]], 'friends': [], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_5.1_4.2-a', 'lambdas': [[5, 1], [5, 1], [4, 2]], 'map': '(1/61509375*(-348496*nu^3-741120*nu^2-220032*nu+1325056)*x^2+1/1121008359375*(-1779172576*nu^3-7720954976*nu^2-13551312128*nu-8754475264)*x+1/6810125783203125*(2876788492172*nu^3-5968135269264*nu^2-36027973300704*nu-60005055586688))/(x^6+1/12150*(-478*nu^3-575*nu^2+880*nu-3124)*x^5+1/59049000*(-220348*nu^3+430695*nu^2+3479136*nu+9690880)*x^4+1/1614252037500*(-1960623166*nu^3-5800744319*nu^2-10636114256*nu-1393134520)*x^3+1/26150883007500000*(-16499085466792*nu^3+36659243012625*nu^2+213540923222592*nu+352327034626936)*x^2+1/2382999214058437500000*(811866697332031226*nu^3+1370168723740622515*nu^2-538453590192224528*nu-4790319774148953016)*x+1/173720642704860093750000000*(-321238464333470237716*nu^3-4302581215060946019141*nu^2-11678312737214932209888*nu-13146805620882796103864))*y+(1/61509375*(348496*nu^3+741120*nu^2+220032*nu-1325056)*x^3+1/373669453125*(-469645192*nu^3+437700016*nu^2+4180837696*nu+7636429952)*x^2+1/11350209638671875*(-852829887316*nu^3+1270879441200*nu^2+9158542172448*nu+15867682562176)*x+1/620572711994384765625*(-79784546770981598*nu^3-156472493857022308*nu^2-16062883459261264*nu+383429587505849248))/(x^6+1/12150*(-478*nu^3-575*nu^2+880*nu-3124)*x^5+1/59049000*(-220348*nu^3+430695*nu^2+3479136*nu+9690880)*x^4+1/1614252037500*(-1960623166*nu^3-5800744319*nu^2-10636114256*nu-1393134520)*x^3+1/26150883007500000*(-16499085466792*nu^3+36659243012625*nu^2+213540923222592*nu+352327034626936)*x^2+1/2382999214058437500000*(811866697332031226*nu^3+1370168723740622515*nu^2-538453590192224528*nu-4790319774148953016)*x+1/173720642704860093750000000*(-321238464333470237716*nu^3-4302581215060946019141*nu^2-11678312737214932209888*nu-13146805620882796103864))', 'orbit_size': 4, 'pass_size': 8, 'plabel': '6T15-5.1_5.1_4.2', 'plane_constant': '1/36*(-25*nu^2 + 25)', 'plane_map_constant_factored': '\\frac{2^{2} \\cdot 3}{5^{3}} (\\nu^{2} - 1)', 'plane_model': '(-2)*t^2 - (-18*x^4 + (3*nu^3 + 12*nu^2 - 30*nu + 16)*x^3 + (12*nu^3 - 23*nu^2 + 48*nu - 7)*x^2 + 1/4*(-69*nu^3 + 140*nu^2 + 42*nu - 272)*x + 1/2*(12*nu^3 - 13*nu^2 - 32*nu + 128))*t + x^5*(x + 1/4*(-3*nu^3 - 4*nu^2 + 6*nu + 16))', 'plane_model_latex': '\\left(-2\\right) t^{2} + \\left(18 x^{4} + \\left(-3 \\nu^{3} - 12 \\nu^{2} + 30 \\nu - 16\\right) x^{3} + \\left(-12 \\nu^{3} + 23 \\nu^{2} - 48 \\nu + 7\\right) x^{2} + 1/4 \\left(69 \\nu^{3} - 140 \\nu^{2} - 42 \\nu + 272\\right) x + 1/2 \\left(-12 \\nu^{3} + 13 \\nu^{2} + 32 \\nu - 128\\right)\\right) t + x^{5} \\left(x + 1/4 \\left(-3 \\nu^{3} - 4 \\nu^{2} + 6 \\nu + 16\\right)\\right)', 'primitivization': '6T15-5.1_5.1_4.2-a', 'triples': [[[3, 1, 4, 5, 2, 6], [3, 2, 5, 1, 6, 4], [2, 3, 4, 1, 6, 5]], [[2, 4, 3, 5, 6, 1], [2, 6, 1, 3, 5, 4], [2, 3, 4, 1, 6, 5]], [[3, 2, 4, 5, 6, 1], [3, 6, 2, 1, 5, 4], [2, 3, 4, 1, 6, 5]], [[2, 4, 5, 3, 1, 6], [2, 5, 1, 4, 6, 3], [2, 3, 4, 1, 6, 5]]], 'triples_cyc': [['(1,3,4,5,2)', '(1,3,5,6,4)', '(1,2,3,4)(5,6)'], ['(1,2,4,5,6)', '(1,2,6,4,3)', '(1,2,3,4)(5,6)'], ['(1,3,4,5,6)', '(1,3,2,6,4)', '(1,2,3,4)(5,6)'], ['(1,2,4,3,5)', '(1,2,5,6,3)', '(1,2,3,4)(5,6)']]}
91. label: 6T15-5.1_5.1_4.2-b
    {'BelyiDB_label': '6T15-[5,5,4]-51-51-42-g1-b', 'BelyiDB_plabel': '6T15-[5,5,4]-51-51-42-g1', 'a_s': 4, 'abc': [5, 5, 4], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [16, -8, -1, -2, 1], 'base_field_label': '4.2.24000.2', 'c_s': 5, 'curve': 'y^2=x^3+1/1500*(461873*nu^3+303251*nu^2+343853*nu-2781873)*x+1/675000*(13006663061*nu^3+8539985708*nu^2+9680539295*nu-78336123534)', 'deg': 6, 'embeddings': [[1.505692194778891, 0.0], [-1.08113883008419, 1.68259883218912], [-1.08113883008419, -1.68259883218912], [2.656585465389488, 0.0]], 'friends': [], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_5.1_4.2-b', 'lambdas': [[5, 1], [5, 1], [4, 2]], 'map': '(1/3125*(-31527800*nu^3-20700720*nu^2-23465160*nu+189884384)*x^2+1/234375*(667862685584*nu^3+438508932960*nu^2+497073766512*nu-4022382527744)*x+1/3515625*(-716301533739154*nu^3-470313175889460*nu^2-533125613509662*nu+4314118513837096))/(x^6+1/50*(-1131*nu^3-740*nu^2-833*nu+6766)*x^5+1/1500*(5443267*nu^3+3573921*nu^2+4051335*nu-32783419)*x^4+1/337500*(-91020619129*nu^3-59762809996*nu^2-67744454251*nu+548196201750)*x^3+1/6750000*(68843774109252*nu^3+45201821455427*nu^2+51238728028718*nu-414630133672123)*x^2+1/1012500000*(-194084609998403927*nu^3-127433133986179752*nu^2-144452401534186749*nu+1168926729602978414)*x+1/91125000000*(129935425649034842257*nu^3+85313711920499276485*nu^2+96707741436728541721*nu-782571024900021213807))*y+(1/3125*(31527800*nu^3+20700720*nu^2+23465160*nu-189884384)*x^3+1/78125*(-89049596772*nu^3-58468670984*nu^2-66277423292*nu+536325129328)*x^2+1/5859375*(-5512733509739330*nu^3-3619580697094804*nu^2-4102991960818414*nu+33201919270033768)*x+1/1318359375*(165864531274466967767*nu^3+108904240458457394862*nu^2+123448891046743261569*nu-998963720522495422964))/(x^6+1/50*(-1131*nu^3-740*nu^2-833*nu+6766)*x^5+1/1500*(5443267*nu^3+3573921*nu^2+4051335*nu-32783419)*x^4+1/337500*(-91020619129*nu^3-59762809996*nu^2-67744454251*nu+548196201750)*x^3+1/6750000*(68843774109252*nu^3+45201821455427*nu^2+51238728028718*nu-414630133672123)*x^2+1/1012500000*(-194084609998403927*nu^3-127433133986179752*nu^2-144452401534186749*nu+1168926729602978414)*x+1/91125000000*(129935425649034842257*nu^3+85313711920499276485*nu^2+96707741436728541721*nu-782571024900021213807))', 'orbit_size': 4, 'pass_size': 8, 'plabel': '6T15-5.1_5.1_4.2', 'plane_constant': '1/9*(130*nu^3 - 10*nu^2 - 110*nu - 1165)', 'plane_map_constant_factored': '\\frac{1}{5^{2}} (6 \\nu^{3} - 34 \\nu^{2} + 62 \\nu - 37)', 'plane_model': '(1/4*(-13*nu^3 + 10*nu^2 + 17*nu + 128))*t^2 - (1/4*(nu^3 - 2*nu^2 - 5*nu)*x^4 + (-3*nu^3 + 2*nu^2 + 7*nu + 32)*x^3 + 1/4*(103*nu^3 - 54*nu^2 - 187*nu - 1104)*x^2 + 1/2*(-99*nu^3 + 50*nu^2 + 175*nu + 1056)*x + 24*nu^3 - 12*nu^2 - 42*nu - 256)*t + (1/4*(-3*nu^3 - 2*nu^2 - nu + 16))*x^5*(x + 8*nu^3 - 4*nu^2 - 14*nu - 86)', 'plane_model_latex': '\\left(1/4 \\left(-13 \\nu^{3} + 10 \\nu^{2} + 17 \\nu + 128\\right)\\right) t^{2} - \\left(1/4 \\left(\\nu^{3} - 2 \\nu^{2} - 5 \\nu\\right) x^{4} + \\left(-3 \\nu^{3} + 2 \\nu^{2} + 7 \\nu + 32\\right) x^{3} + 1/4 \\left(103 \\nu^{3} - 54 \\nu^{2} - 187 \\nu - 1104\\right) x^{2} + 1/2 \\left(-99 \\nu^{3} + 50 \\nu^{2} + 175 \\nu + 1056\\right) x + 24 \\nu^{3} - 12 \\nu^{2} - 42 \\nu - 256\\right) t + \\left(1/4 \\left(-3 \\nu^{3} - 2 \\nu^{2} - \\nu + 16\\right)\\right) x^{5} \\left(x + 8 \\nu^{3} - 4 \\nu^{2} - 14 \\nu - 86\\right)', 'primitivization': '6T15-5.1_5.1_4.2-b', 'triples': [[[2, 3, 5, 4, 6, 1], [4, 6, 1, 2, 5, 3], [2, 3, 4, 1, 6, 5]], [[5, 1, 3, 6, 4, 2], [5, 2, 6, 3, 4, 1], [2, 3, 4, 1, 6, 5]], [[4, 6, 3, 5, 2, 1], [1, 6, 5, 3, 2, 4], [2, 3, 4, 1, 6, 5]], [[3, 4, 2, 5, 1, 6], [2, 5, 3, 1, 6, 4], [2, 3, 4, 1, 6, 5]]], 'triples_cyc': [['(1,2,3,5,6)', '(1,4,2,6,3)', '(1,2,3,4)(5,6)'], ['(1,5,4,6,2)', '(1,5,4,3,6)', '(1,2,3,4)(5,6)'], ['(1,4,5,2,6)', '(2,6,4,3,5)', '(1,2,3,4)(5,6)'], ['(1,3,2,4,5)', '(1,2,5,6,4)', '(1,2,3,4)(5,6)']]}
92. label: 6T15-5.1_5.1_5.1-a
    {'BelyiDB_label': '6T15-[5,5,5]-51-51-51-g1-a', 'BelyiDB_plabel': '6T15-[5,5,5]-51-51-51-g1', 'a_s': 5, 'abc': [5, 5, 5], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [4, -1, 1], 'base_field_label': '2.0.15.1', 'c_s': 5, 'curve': 'y^2=x^3+1/1586874322944*(642624002700*nu+536036017025)*x+1/8995520821969944576*(-6936979531903347500*nu+8175151993626563125)', 'deg': 6, 'embeddings': [[0.5, 1.936491673103708], [0.5, -1.936491673103708]], 'friends': [], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_5.1_5.1-a', 'lambdas': [[5, 1], [5, 1], [5, 1]], 'map': '(1/2380311484416*(-2193194782050*nu+20956699971025)*x^2+1/4497760410984972288*(-6090283026038501875*nu+146847217947822725000)*x+1/11331765525685402821722112*(-87114146698867485378000000*nu+419942860267513847830740625))/(x^6+1/1259712*(1108255*nu+7557010)*x^5+1/1586874322944*(4602047896125*nu+21088471244375)*x^4+1/8995520821969944576*(34397745373162311875*nu+132253357580239358750)*x^3+1/7554510350456935214481408*(19044515385215138317668750*nu+66313073481285951407628125)*x^2+1/28549522027784420318714410303488*(23723962963730926563126920921875*nu+77556637789595651739658726581250)*x+1/107892526477993103065585101696682426368*(11821139854834880270032018801220671875*nu+37015024095001253614157938082993921875))*y+(1/11331765525685402821722112*(87114146698867485378000000*nu-419942860267513847830740625)*x+1/42824283041676630478071615455232*(876968261926129614623426256390625*nu-1416944073978653128685659599406250))/(x^6+1/1259712*(1108255*nu+7557010)*x^5+1/1586874322944*(4602047896125*nu+21088471244375)*x^4+1/8995520821969944576*(34397745373162311875*nu+132253357580239358750)*x^3+1/7554510350456935214481408*(19044515385215138317668750*nu+66313073481285951407628125)*x^2+1/28549522027784420318714410303488*(23723962963730926563126920921875*nu+77556637789595651739658726581250)*x+1/107892526477993103065585101696682426368*(11821139854834880270032018801220671875*nu+37015024095001253614157938082993921875))', 'orbit_size': 2, 'pass_size': 8, 'plabel': '6T15-5.1_5.1_5.1', 'plane_constant': '1/144*(-5*nu + 4)', 'plane_map_constant_factored': '\\frac{3}{2} (5 \\nu - 1)', 'plane_model': '(-3*nu - 9)*x^5*((nu + 1)*x + 903*nu + 768)*t^2 + (1/3*(-nu + 2))*((nu + 1)*x^6 + (903*nu + 768)*x^5 + (-2123048161161*nu + 33827052851334)*x - 1012675229664858*nu + 2337426205996032)*t + (1/3*(317761641280*nu - 27289133717))*(6*x - 410*nu - 1607)', 'plane_model_latex': '\\left(-3 \\nu - 9\\right) x^{5} \\left(\\left(\\nu + 1\\right) x + 903 \\nu + 768\\right) t^{2} + \\left(1/3 \\left(-\\nu + 2\\right)\\right) \\left(\\left(\\nu + 1\\right) x^{6} + \\left(903 \\nu + 768\\right) x^{5} + \\left(-2123048161161 \\nu + 33827052851334\\right) x - 1012675229664858 \\nu + 2337426205996032\\right) t + \\left(1/3 \\left(317761641280 \\nu - 27289133717\\right)\\right) \\left(6 x - 410 \\nu - 1607\\right)', 'primitivization': '6T15-5.1_5.1_5.1-a', 'triples': [[[2, 3, 4, 5, 1, 6], [3, 1, 5, 4, 6, 2], [3, 2, 6, 1, 4, 5]], [[2, 3, 4, 5, 1, 6], [3, 2, 5, 6, 4, 1], [3, 6, 2, 1, 5, 4]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,3,5,6,2)', '(1,3,6,5,4)'], ['(1,2,3,4,5)', '(1,3,5,4,6)', '(1,3,2,6,4)']]}
93. label: 6T15-5.1_5.1_5.1-b
    {'BelyiDB_label': '6T15-[5,5,5]-51-51-51-g1-b', 'BelyiDB_plabel': '6T15-[5,5,5]-51-51-51-g1', 'a_s': 5, 'abc': [5, 5, 5], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [4, -1, 1], 'base_field_label': '2.0.15.1', 'c_s': 5, 'curve': 'y^2=x^3+1/160000*(-657*nu-4419)*x+1/6400000*(20013*nu-54057)', 'deg': 6, 'embeddings': [[0.5, -1.936491673103708], [0.5, 1.936491673103708]], 'friends': [], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_5.1_5.1-b', 'lambdas': [[5, 1], [5, 1], [5, 1]], 'map': '(1/200000*(-11745*nu-5427)*x^2+1/625000*(-19197*nu-22599)*x+1/32000000000*(-12725667*nu-360573849))/(x^6+1/200*(-21*nu+33)*x^5+1/32000*(-1935*nu+10467)*x^4+1/3200000*(106701*nu+155319)*x^3+1/5120000000*(28837917*nu-38227545)*x^2+1/5120000000000*(-70827885*nu-3742303383)*x+1/4096000000000000*(-36228501693*nu-5590861191))*y+(1/200000*(11745*nu+5427)*x^3+1/80000000*(26973*nu+1165671)*x^2+1/160000000000*(-668868435*nu+978410583)*x+1/64000000000000*(-35387022501*nu-3000230847))/(x^6+1/200*(-21*nu+33)*x^5+1/32000*(-1935*nu+10467)*x^4+1/3200000*(106701*nu+155319)*x^3+1/5120000000*(28837917*nu-38227545)*x^2+1/5120000000000*(-70827885*nu-3742303383)*x+1/4096000000000000*(-36228501693*nu-5590861191))', 'orbit_size': 2, 'pass_size': 8, 'plabel': '6T15-5.1_5.1_5.1', 'plane_constant': '1/2*(nu + 1)', 'plane_map_constant_factored': '\\frac{1}{3} (-\\nu + 2)', 'plane_model': '(-3)*x^5*((-nu + 1)*x - nu + 2)*t^2 + 1/2*nu*(6*x - nu + 1)*t + (1/2*(-nu + 2))*(6*x - nu + 1)', 'plane_model_latex': '\\left(-3\\right) x^{5} \\left(\\left(-\\nu + 1\\right) x - \\nu + 2\\right) t^{2} + 1/2 \\nu \\left(6 x - \\nu + 1\\right) t + \\left(1/2 \\left(-\\nu + 2\\right)\\right) \\left(6 x - \\nu + 1\\right)', 'primitivization': '6T15-5.1_5.1_5.1-b', 'triples': [[[2, 3, 4, 5, 1, 6], [3, 1, 4, 6, 5, 2], [5, 2, 6, 1, 3, 4]], [[2, 3, 4, 5, 1, 6], [1, 3, 5, 6, 4, 2], [3, 1, 6, 2, 5, 4]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,3,4,6,2)', '(1,5,3,6,4)'], ['(1,2,3,4,5)', '(2,3,5,4,6)', '(1,3,6,4,2)']]}
94. label: 6T15-5.1_5.1_5.1-c
    {'BelyiDB_label': '6T15-[5,5,5]-51-51-51-g1-c', 'BelyiDB_plabel': '6T15-[5,5,5]-51-51-51-g1', 'a_s': 5, 'abc': [5, 5, 5], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [1, -1, 1], 'base_field_label': '2.0.3.1', 'c_s': 5, 'curve': 'y^2=x^3-27/12500', 'curve_label': '2.0.3.1-5625.1-CMa1', 'deg': 6, 'embeddings': [[0.5, 0.8660254037844387], [0.5, -0.8660254037844387]], 'friends': ['EllipticCurve/2.0.3.1/5625.1/CMa/1'], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_5.1_5.1-c', 'lambdas': [[5, 1], [5, 1], [5, 1]], 'map': '(1/3125*(162*nu-81)*x^2+1/78125*(972*nu-486)*x+1/390625*(-1458*nu+729))/(x^6-9/25*x^5+27/125*x^4+1/3125*(-162*nu-54)*x^3+1/78125*(729*nu-486)*x^2+1/9765625*(-2187*nu+5832)*x+1/244140625*(-2187*nu-1458))*y+(1/3125*(-162*nu+81)*x^3+1/156250*(1458*nu-729)*x^2+1/9765625*(-2187*nu+10935)*x+1/488281250*(-4374*nu-76545))/(x^6-9/25*x^5+27/125*x^4+1/3125*(-162*nu-54)*x^3+1/78125*(729*nu-486)*x^2+1/9765625*(-2187*nu+5832)*x+1/244140625*(-2187*nu-1458))', 'orbit_size': 2, 'pass_size': 8, 'plabel': '6T15-5.1_5.1_5.1', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': 'x^5*(x + 3)*t^2 + (x^6 + 3*x^5 + (-18*nu + 9)*x + 6*nu - 3)*t + (3*nu + 3)*((-nu - 1)*x + 2*nu - 3)', 'plane_model_latex': 'x^{5} \\left(x + 3\\right) t^{2} + \\left(x^{6} + 3 x^{5} + \\left(-18 \\nu + 9\\right) x + 6 \\nu - 3\\right) t - \\left(6 \\nu - 3\\right) \\left(\\left(-\\nu + 2\\right) x - 3 \\nu + 1\\right)', 'primitivization': '6T15-5.1_5.1_5.1-c', 'triples': [[[2, 3, 4, 5, 1, 6], [2, 5, 3, 6, 4, 1], [2, 6, 1, 3, 5, 4]], [[2, 3, 4, 5, 1, 6], [4, 1, 3, 5, 6, 2], [4, 2, 6, 3, 1, 5]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,2,5,4,6)', '(1,2,6,4,3)'], ['(1,2,3,4,5)', '(1,4,5,6,2)', '(1,4,3,6,5)']]}
95. label: 6T15-5.1_5.1_5.1-d
    {'BelyiDB_label': '6T15-[5,5,5]-51-51-51-g1-d', 'BelyiDB_plabel': '6T15-[5,5,5]-51-51-51-g1', 'a_s': 5, 'abc': [5, 5, 5], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 5, 'base_field': [4, -1, 1], 'base_field_label': '2.0.15.1', 'c_s': 5, 'curve': 'y^2=x^3+1/160000*(-657*nu-4419)*x+1/6400000*(20013*nu-54057)', 'deg': 6, 'embeddings': [[0.5, 1.936491673103708], [0.5, -1.936491673103708]], 'friends': [], 'g': 1, 'geomtype': 'H', 'group': '6T15', 'group_num': 15, 'is_primitive': True, 'label': '6T15-5.1_5.1_5.1-d', 'lambdas': [[5, 1], [5, 1], [5, 1]], 'map': '(1/200000*(11745*nu+5427)*x^2+1/625000*(19197*nu+22599)*x+1/32000000000*(12725667*nu+360573849))/(x^6+1/200*(-21*nu+33)*x^5+1/32000*(-1935*nu+10467)*x^4+1/3200000*(-269139*nu-18345)*x^3+1/5120000000*(25385373*nu-187433433)*x^2+1/5120000000000*(8762319315*nu+497373993)*x+1/4096000000000000*(-79212394557*nu+549870373881))*y+(1/200000*(-11745*nu-5427)*x^3+1/80000000*(-26973*nu-1165671)*x^2+1/160000000000*(-392832585*nu+1110900501)*x+1/64000000000000*(-36058645827*nu+5678850951))/(x^6+1/200*(-21*nu+33)*x^5+1/32000*(-1935*nu+10467)*x^4+1/3200000*(-269139*nu-18345)*x^3+1/5120000000*(25385373*nu-187433433)*x^2+1/5120000000000*(8762319315*nu+497373993)*x+1/4096000000000000*(-79212394557*nu+549870373881))', 'orbit_size': 2, 'pass_size': 8, 'plabel': '6T15-5.1_5.1_5.1', 'plane_constant': '1/2*(nu + 1)', 'plane_map_constant_factored': '\\frac{1}{3} (-\\nu + 2)', 'plane_model': '(-3)*x^5*((-nu + 1)*x - nu + 2)*t^2 - 1/2*nu*(6*x - nu + 1)*t + (1/2*(-nu + 2))*(6*x - nu + 1)', 'plane_model_latex': '\\left(-3\\right) x^{5} \\left(\\left(-\\nu + 1\\right) x - \\nu + 2\\right) t^{2} - 1/2 \\nu \\left(6 x - \\nu + 1\\right) t + \\left(1/2 \\left(-\\nu + 2\\right)\\right) \\left(6 x - \\nu + 1\\right)', 'primitivization': '6T15-5.1_5.1_5.1-d', 'triples': [[[2, 3, 4, 5, 1, 6], [4, 2, 5, 3, 6, 1], [3, 6, 2, 4, 1, 5]], [[2, 3, 4, 5, 1, 6], [4, 1, 3, 6, 2, 5], [6, 2, 5, 3, 1, 4]]], 'triples_cyc': [['(1,2,3,4,5)', '(1,4,3,5,6)', '(1,3,2,6,5)'], ['(1,2,3,4,5)', '(1,4,6,5,2)', '(1,6,4,3,5)']]}
96. label: 6T16-3.2.1_3.2.1_3.3-a
    {'BelyiDB_label': '6T16-[6,6,3]-321-321-33-g0-a', 'BelyiDB_plabel': '6T16-[6,6,3]-321-321-33-g0', 'a_s': 3, 'abc': [6, 6, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 6, 'base_field': [-2, 0, 0, 1], 'base_field_label': '3.1.108.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[-0.6299605249474366, 1.091123635971721], [-0.6299605249474366, -1.091123635971721], [1.259921049894873, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T16', 'group_num': 16, 'is_primitive': True, 'label': '6T16-3.2.1_3.2.1_3.3-a', 'lambdas': [[3, 2, 1], [3, 2, 1], [3, 3]], 'map': '(1/22665187*(47845344*nu^2+124919136*nu-187935504)*x^6+1/22665187*(161863128*nu^2+320599920*nu-645759312)*x^5+1/22665187*(184953312*nu^2+295902720*nu-672005568)*x^4+1/22665187*(61651104*nu^2+98634240*nu-224001856)*x^3)/(x^6+1/283*(228*nu^2+192*nu-732)*x^5+1/80089*(-34452*nu^2-4704*nu+29820)*x^4+1/22665187*(-43226496*nu^2-28574016*nu+107640032)*x^3+1/22665187*(10371552*nu^2-2180448*nu-12400944)*x^2+1/22665187*(38381760*nu^2+14527872*nu-79465152)*x+1/22665187*(12793920*nu^2+4842624*nu-26488384))', 'orbit_size': 3, 'pass_size': 3, 'plabel': '6T16-3.2.1_3.2.1_3.3', 'plane_constant': '1', 'plane_map_constant_factored': '', 'plane_model': '(-nu + 1)*(nu^2*x - nu^2 - nu - 1)*((nu^2 + 1)*x + 1)^2*t + x^3*(x - nu^2 - 1)^2*(x - nu^2 + 2)', 'plane_model_latex': '\\left(-\\nu + 1\\right) \\left(\\nu^{2} x - \\nu^{2} - \\nu - 1\\right) \\left(\\left(\\nu^{2} + 1\\right) x + 1\\right)^{2} t + x^{3} \\left(x - \\nu^{2} - 1\\right)^{2} \\left(x - \\nu^{2} + 2\\right)', 'primitivization': '6T16-3.2.1_3.2.1_3.3-a', 'triples': [[[2, 3, 1, 5, 4, 6], [5, 2, 4, 6, 1, 3], [6, 5, 2, 1, 3, 4]], [[2, 3, 1, 5, 4, 6], [4, 6, 3, 1, 2, 5], [3, 4, 5, 6, 1, 2]], [[2, 3, 1, 5, 4, 6], [2, 4, 6, 1, 5, 3], [6, 4, 1, 5, 2, 3]]], 'triples_cyc': [['(1,2,3)(4,5)', '(1,5)(3,4,6)', '(1,6,4)(2,5,3)'], ['(1,2,3)(4,5)', '(1,4)(2,6,5)', '(1,3,5)(2,4,6)'], ['(1,2,3)(4,5)', '(1,2,4)(3,6)', '(1,6,3)(2,4,5)']]}
97. label: 6T16-4.1.1_3.2.1_3.3-a
    {'BelyiDB_label': '6T16-[4,6,3]-411-321-33-g0-a', 'BelyiDB_plabel': '6T16-[4,6,3]-411-321-33-g0', 'a_s': 3, 'abc': [4, 6, 3], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-1, 1], 'base_field_label': '1.1.1.1', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[1.0, 0.0]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T16', 'group_num': 16, 'is_primitive': True, 'label': '6T16-4.1.1_3.2.1_3.3-a', 'lambdas': [[4, 1, 1], [3, 2, 1], [3, 3]], 'map': '(140625/137842*x^6-18750/68921*x^5-6250/68921*x^4)/(x^6-8/41*x^5-920/5043*x^4+46720/1860867*x^3+7360/620289*x^2-512/620289*x-512/1860867)', 'orbit_size': 1, 'pass_size': 1, 'plabel': '6T16-4.1.1_3.2.1_3.3', 'plane_constant': '1/27', 'plane_map_constant_factored': '3^{3}', 'plane_model': '(-5*x + 1)^2*(4*x + 1)*t - x^4*(-27*x^2 - 18*x + 5)', 'plane_model_latex': '\\left(-5 x + 1\\right)^{2} \\left(4 x + 1\\right) t - x^{4} \\left(-27 x^{2} - 18 x + 5\\right)', 'primitivization': '6T16-4.1.1_3.2.1_3.3-a', 'triples': [[[2, 3, 4, 1, 5, 6], [6, 4, 3, 5, 2, 1], [2, 6, 5, 3, 4, 1]]], 'triples_cyc': [['(1,2,3,4)', '(1,6)(2,4,5)', '(1,2,6)(3,5,4)']]}
98. label: 6T16-4.2_3.2.1_3.2.1-a
    {'BelyiDB_label': '6T16-[4,6,6]-42-321-321-g0-a', 'BelyiDB_plabel': '6T16-[4,6,6]-42-321-321-g0', 'a_s': 4, 'abc': [4, 6, 6], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 6, 'base_field': [-2, 4, -3, -2, 1], 'base_field_label': '4.2.8640.2', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[-1.667830653621998, 0.0], [2.667830653621998, 0.0], [0.5, -0.4466427462560857], [0.5, 0.4466427462560857]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T16', 'group_num': 16, 'is_primitive': True, 'label': '6T16-4.2_3.2.1_3.2.1-a', 'lambdas': [[4, 2], [3, 2, 1], [3, 2, 1]], 'map': '(1/2575229264248*(-943419815504262*nu^3+3462885362098539*nu^2-2948961730538520*nu+1136913428234853)*x^6+1/439608956*(361186011198*nu^3-1322636947929*nu^2+1121188755528*nu-434752834059)*x^5+1/879217912*(-410597765334*nu^3+1489954407903*nu^2-1227493417320*nu+467508352941)*x^4)/(x^6+1/479*(8112*nu^3-28827*nu^2+23733*nu-11226)*x^5+1/458882*(-679757016*nu^3+2488087746*nu^2-2104686054*nu+812483505)*x^4+1/109902239*(356081692791*nu^3-1303834976757*nu^2+1102863151665*nu-424363392899)*x^3+1/109902239*(-37727981691*nu^3+137306715363*nu^2-113335505721*nu+41408461563)*x^2+1/109902239*(-357278717559*nu^3+1309233839940*nu^2-1110297056382*nu+428370381765)*x+1/219804478*(402387975294*nu^3-1468837926108*nu^2+1230838468380*nu-469353758099))', 'orbit_size': 4, 'pass_size': 4, 'plabel': '6T16-4.2_3.2.1_3.2.1', 'plane_constant': '1/375*(-48*nu^3 + 87*nu^2 + 200*nu - 202)', 'plane_map_constant_factored': '\\frac{1}{2^{4}} (26 \\nu^{3} - 27 \\nu^{2} - 60 \\nu + 11)', 'plane_model': '(4*nu^3 - 15*nu^2 + 13*nu - 5)*x^3*((-2*nu^3 + 3*nu^2 + 8*nu - 4)*x + nu^3 - 3*nu^2 - 4*nu + 7)^2*(x + nu^2 - 2*nu - 1)*t + (nu^2 + nu - 1)*((nu^3 - 3*nu^2 + 1)*x - 1)^2', 'plane_model_latex': '\\left(4 \\nu^{3} - 15 \\nu^{2} + 13 \\nu - 5\\right) x^{3} \\left(\\left(-2 \\nu^{3} + 3 \\nu^{2} + 8 \\nu - 4\\right) x + \\nu^{3} - 3 \\nu^{2} - 4 \\nu + 7\\right)^{2} \\left(x + \\nu^{2} - 2 \\nu - 1\\right) t + \\left(\\nu^{2} + \\nu - 1\\right) \\left(\\left(\\nu^{3} - 3 \\nu^{2} + 1\\right) x - 1\\right)^{2}', 'primitivization': '6T16-4.2_3.2.1_3.2.1-a', 'triples': [[[4, 6, 1, 5, 3, 2], [2, 3, 1, 5, 4, 6], [2, 6, 4, 3, 5, 1]], [[2, 4, 1, 3, 6, 5], [5, 3, 2, 4, 6, 1], [2, 6, 4, 3, 5, 1]], [[4, 1, 6, 5, 2, 3], [2, 6, 4, 3, 5, 1], [1, 5, 2, 6, 3, 4]], [[3, 5, 6, 1, 2, 4], [2, 3, 1, 5, 4, 6], [5, 4, 3, 6, 1, 2]]], 'triples_cyc': [['(1,4,5,3)(2,6)', '(1,2,3)(4,5)', '(1,2,6)(3,4)'], ['(1,2,4,3)(5,6)', '(1,5,6)(2,3)', '(1,2,6)(3,4)'], ['(1,4,5,2)(3,6)', '(1,2,6)(3,4)', '(2,5,3)(4,6)'], ['(1,3,6,4)(2,5)', '(1,2,3)(4,5)', '(1,5)(2,4,6)']]}
99. label: 6T16-4.2_3.2.1_4.1.1-a
    {'BelyiDB_label': '6T16-[4,6,4]-42-321-411-g0-a', 'BelyiDB_plabel': '6T16-[4,6,4]-42-321-411-g0', 'a_s': 4, 'abc': [4, 6, 4], 'aut_group': [[1, 2, 3, 4, 5, 6]], 'b_s': 4, 'base_field': [-6, 0, 6, -2, 1], 'base_field_label': '4.2.8640.3', 'c_s': 6, 'curve': 'PP1', 'deg': 6, 'embeddings': [[-0.8449593779511566, 0.0], [0.875245285186537, 2.392062597386565], [1.094468807578083, 0.0], [0.875245285186537, -2.392062597386565]], 'friends': [], 'g': 0, 'geomtype': 'H', 'group': '6T16', 'group_num': 16, 'is_primitive': True, 'label': '6T16-4.2_3.2.1_4.1.1-a', 'lambdas': [[4, 2], [3, 2, 1], [4, 1, 1]], 'map': '(1/39217501119604*(111156951742290*nu^3-127159990092252*nu^2+635883676109532*nu+738902108265417)*x^6+1/9804375279901*(5047261797474*nu^3-17962902488250*nu^2-12117812791428*nu-28998026475039)*x^5+1/9804375279901*(-1478060873334*nu^3+13548404268144*nu^2-14870378462436*nu+4271441010705)*x^4)/(x^6+1/2903394255*(1607968064*nu^3-2856739716*nu^2+4816089336*nu-5746773612)*x^5+1/21485117487*(-14390212036*nu^3+51264888372*nu^2-26384686800*nu-10616639376)*x^4+1/794949347019*(-587615002448*nu^3+41985816120*nu^2-3063614902464*nu+4061778224208)*x^3+1/29413125839703*(4438357765312*nu^3-9174225945384*nu^2-15910633518240*nu+22598278667652)*x^2+1/29413125839703*(64435102203200*nu^3-168346797176448*nu^2+473565337166208*nu-401121851338992)*x+1/29413125839703*(-46731576315584*nu^3+136938630610080*nu^2-379853700172032*nu+312970587878544))', 'orbit_size': 4, 'pass_size': 4, 'plabel': '6T16-4.2_3.2.1_4.1.1', 'plane_constant': '1/625*(99*nu^3 - 216*nu^2 + 531*nu - 1017)', 'plane_map_constant_factored': '\\frac{1}{3^{3}} (-2 \\nu^{3} + 3 \\nu^{2} - 9 \\nu - 21)', 'plane_model': '(1/5*(-11*nu^3 + 79*nu^2 + 161*nu + 73))*((nu^3 - 2*nu^2 + 8*nu - 3)*x^2 + 1/5*(49972*nu^3 - 142168*nu^2 + 419958*nu - 354846)*x + 1/5*(198264737*nu^3 - 564055123*nu^2 + 1666192088*nu - 1407864631))*t + (1/5*(-4887005488*nu^3 - 5018038448*nu^2 + 6075705248*nu + 5768229509))*x^4*(x + 1/5*(38093*nu^3 - 108372*nu^2 + 320127*nu - 270489))^2', 'plane_model_latex': '\\left(1/5 \\left(11 \\nu^{3} - 79 \\nu^{2} - 161 \\nu - 73\\right)\\right) \\left(\\left(-\\nu^{3} + 2 \\nu^{2} - 8 \\nu + 3\\right) x^{2} + 1/5 \\left(-49972 \\nu^{3} + 142168 \\nu^{2} - 419958 \\nu + 354846\\right) x + 1/5 \\left(-198264737 \\nu^{3} + 564055123 \\nu^{2} - 1666192088 \\nu + 1407864631\\right)\\right) t + \\left(1/5 \\left(-4887005488 \\nu^{3} - 5018038448 \\nu^{2} + 6075705248 \\nu + 5768229509\\right)\\right) x^{4} \\left(x + 1/5 \\left(38093 \\nu^{3} - 108372 \\nu^{2} + 320127 \\nu - 270489\\right)\\right)^{2}', 'primitivization': '6T16-4.2_3.2.1_4.1.1-a', 'triples': [[[2, 3, 4, 1, 6, 5], [3, 1, 2, 5, 4, 6], [5, 2, 3, 1, 6, 4]], [[2, 3, 4, 1, 6, 5], [5, 3, 2, 4, 6, 1], [4, 6, 3, 2, 5, 1]], [[2, 3, 4, 1, 6, 5], [6, 1, 4, 3, 5, 2], [3, 2, 6, 4, 1, 5]], [[2, 3, 4, 1, 6, 5], [6, 2, 4, 3, 1, 5], [3, 5, 2, 4, 1, 6]]], 'triples_cyc': [['(1,2,3,4)(5,6)', '(1,3,2)(4,5)', '(1,5,6,4)'], ['(1,2,3,4)(5,6)', '(1,5,6)(2,3)', '(1,4,2,6)'], ['(1,2,3,4)(5,6)', '(1,6,2)(3,4)', '(1,3,6,5)'], ['(1,2,3,4)(5,6)', '(1,6,5)(3,4)', '(1,3,2,5)']]}