LMFDB - Elliptic curve data

Formats: - HTML - YAML - JSON - 2025-05-27T17:54:17.352378

Column	Type
Ciso	text
Clabel	text
Cnumber	smallint
abc_quality	double precision
absD	numeric
adelic_genus	integer
adelic_index	integer
adelic_level	bigint
ainvs	numeric[]
analytic_rank	smallint
bad_primes	integer[]
class_deg	smallint
class_size	smallint
cm	smallint
conductor	integer
degree	bigint
elladic_images	text[]
faltings_height	numeric
faltings_index	smallint
faltings_ratio	smallint
iso_nlabel	smallint
isogeny_degrees	smallint[]
jinv	numeric[]
lmfdb_iso	text
lmfdb_label	text
lmfdb_number	smallint
manin_constant	smallint
min_quad_twist_ainvs	numeric[]
min_quad_twist_disc	smallint
modell_images	text[]
modm_images	text[]
nonmax_primes	smallint[]
nonmax_rad	integer
num_bad_primes	smallint
num_int_pts	integer
optimality	smallint
potential_good_reduction	boolean
rank	smallint
regulator	numeric
semistable	boolean
serre_invariants	integer[]
sha	integer
sha_primes	smallint[]
signD	smallint
stable_faltings_height	numeric
szpiro_ratio	double precision
torsion	smallint
torsion_primes	smallint[]
torsion_structure	smallint[]

ec_curvedata • Show schema

{'Ciso': '82110bt', 'Clabel': '82110bt1', 'Cnumber': 1, 'abc_quality': 0.9919472947720096, 'absD': 10116926999101440000, 'adelic_genus': 13, 'adelic_index': 768, 'adelic_level': 1313760, 'ainvs': [1, 0, 0, -49095400, 132402324032], 'analytic_rank': 1, 'bad_primes': [2, 3, 5, 7, 17, 23], 'class_deg': 16, 'class_size': 8, 'cm': 0, 'conductor': 82110, 'degree': 7340032, 'elladic_images': ['16.96.0.95'], 'faltings_height': {'__RealLiteral__': 0, 'data': '2.9543794383492774403065576309', 'prec': 100}, 'faltings_index': 0, 'faltings_ratio': 1, 'iso_nlabel': 44, 'isogeny_degrees': [1, 2, 4, 8, 16], 'jinv': [13087181362848921857775657601, 10116926999101440000], 'lmfdb_iso': '82110.bs', 'lmfdb_label': '82110.bs6', 'lmfdb_number': 6, 'manin_constant': 1, 'min_quad_twist_ainvs': [1, 0, 0, -49095400, 132402324032], 'min_quad_twist_disc': 1, 'modell_images': ['2B'], 'modm_images': ['2.3.0.a.1', '4.12.0-4.c.1.1', '8.48.0-8.bb.1.1', '16.96.0-16.v.1.2', '322.6.0.?', '480.192.1.?', '644.24.0.?', '1288.96.0.?', '2576.192.1.?', '3808.192.2.?', '12512.192.2.?', '28560.192.2.?', '57120.384.5.?', '77280.384.5.?', '87584.384.5.?', '93840.192.2.?', '187680.384.5.?', '656880.384.5.?', '1313760.768.13.?'], 'nonmax_primes': [2], 'nonmax_rad': 2, 'num_bad_primes': 6, 'num_int_pts': 13, 'optimality': 1, 'potential_good_reduction': False, 'rank': 1, 'regulator': {'__RealLiteral__': 0, 'data': '4.0697304843631873930432752506', 'prec': 100}, 'semistable': True, 'serre_invariants': [[2, 4, 161], [3, 4, 27370], [5, 6, 16422], [7, 8, 11730], [17, 18, 4830], [23, 24, 3570]], 'sha': 1, 'sha_primes': [], 'signD': 1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '2.9543794383492774403065576309', 'prec': 100}, 'szpiro_ratio': 5.721322787719762, 'torsion': 8, 'torsion_primes': [2], 'torsion_structure': [8]}

Column	Type
anlist	smallint[]
aplist	smallint[]
class_deg	smallint
class_size	smallint
conductor	integer
isogeny_matrix	smallint[]
lmfdb_iso	text
trace_hash	bigint

ec_classdata • Show schema

{'anlist': [0, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -4, 1, -2, -1, 1, 1, 1, 1, 4, 1], 'aplist': [1, 1, 1, -1, -4, -2, 1, 4, -1, -2, 0, -10, -6, 12, 0, -10, 12, -2, 4, 8, -6, 0, 4, -6, 2], 'class_deg': 16, 'class_size': 8, 'conductor': 82110, 'isogeny_matrix': [[1, 2, 4, 4, 8, 16, 16, 8], [2, 1, 2, 2, 4, 8, 8, 4], [4, 2, 1, 4, 8, 16, 16, 8], [4, 2, 4, 1, 2, 4, 4, 2], [8, 4, 8, 2, 1, 2, 2, 4], [16, 8, 16, 4, 2, 1, 4, 8], [16, 8, 16, 4, 2, 4, 1, 8], [8, 4, 8, 2, 4, 8, 8, 1]], 'lmfdb_iso': '82110.bs', 'trace_hash': 1601533552796115024}

Column	Type
area	numeric
conductor	integer
gens	numeric[]
heights	numeric[]
lmfdb_label	text
ngens	smallint
rank_bounds	smallint[]
real_period	numeric
sha_an	numeric
special_value	numeric
tamagawa_product	integer
torsion_generators	numeric[]
xcoord_integral_points	numeric[]

ec_mwbsd • Show schema

{'area': {'__RealLiteral__': 0, 'data': '0.0027155551355036976658826020275', 'prec': 110}, 'conductor': 82110, 'gens': [[4724, 74468, 1]], 'heights': [{'__RealLiteral__': 0, 'data': '4.0697304843631873930432752506', 'prec': 100}], 'lmfdb_label': '82110.bs6', 'ngens': 1, 'rank_bounds': [1, 1], 'real_period': {'__RealLiteral__': 0, 'data': '0.19044923924242979214769477659', 'prec': 103}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '12.401233194699109664652645957408794859', 'prec': 130}, 'tamagawa_product': 1024, 'torsion_generators': [[4064, 8, 1]], 'xcoord_integral_points': [4048, 4064, 4304, 4724, 5492, 8144, 54224]}

Column	Type
iwdata	jsonb
iwp0	smallint
lmfdb_label	text

ec_iwasawa • Show schema

{'iwdata': {'2': [10, 0], '3': [2, 0], '5': [2, 0], '7': [1, 0], '11': [1, 0], '13': [1, 0], '17': [2, 0], '19': [1, 0], '23': [1, 0], '29': [1, 0], '31': [1, 1, 0], '37': [1, 0], '41': [1, 0], '43': [1, 0], '47': [1, 1, 0]}, 'iwp0': 0, 'lmfdb_label': '82110.bs6'}

Column	Type
degree	smallint
field	numeric[]
lmfdb_label	text
torsion	smallint[]

ec_torsion_growth • Show schema

id: 3860407
{'degree': 2, 'field': [-1466, -1, 1], 'lmfdb_label': '82110.bs6', 'torsion': [16]}
id: 3860408
{'degree': 2, 'field': [-446, -1, 1], 'lmfdb_label': '82110.bs6', 'torsion': [16]}
id: 3860406
{'degree': 2, 'field': [-40, -1, 1], 'lmfdb_label': '82110.bs6', 'torsion': [2, 8]}
id: 3860409
{'degree': 4, 'field': [164836, 0, -973, 0, 1], 'lmfdb_label': '82110.bs6', 'torsion': [2, 16]}
id: 3860414
{'degree': 8, 'field': [-196608, 0, -777232, 0, -1536, 0, 0, 0, 1], 'lmfdb_label': '82110.bs6', 'torsion': [24]}
id: 3860410
{'degree': 8, 'field': [8856576, 3952128, 881792, 60400, -4271, -1410, 2, -2, 1], 'lmfdb_label': '82110.bs6', 'torsion': [4, 8]}
id: 3860413
{'degree': 8, 'field': [1345117456, -71393844, 89318048, -13867335, 610701, 16350, -1438, -3, 1], 'lmfdb_label': '82110.bs6', 'torsion': [32]}
id: 3860411
{'degree': 8, 'field': [11932503696, 918893232, 35380872, 563364, -216791, -8494, 2, -2, 1], 'lmfdb_label': '82110.bs6', 'torsion': [2, 16]}
id: 3860412
{'degree': 8, 'field': [2953552326226, -169626415650, -26707357729, 97851660, 26158281, -11730, -8899, 0, 1], 'lmfdb_label': '82110.bs6', 'torsion': [32]}
id: 3860418
{'degree': 16, 'field': [6894506999808, 7556917690368, 2501445058560, -62671970304, -218228050944, -55942905600, -8169478144, -662868736, 99660736, 30595728, 1427476, -184337, -15832, 273, -73, -4, 1], 'lmfdb_label': '82110.bs6', 'torsion': [2, 24]}
id: 3860420
{'degree': 16, 'field': [34218263912094684900, 0, 455715450355666815, 0, -10659488136489729, 0, -74373235976601, 0, 953496855495, 0, -3263953275, 0, 4935205, 0, -3451, 0, 1], 'lmfdb_label': '82110.bs6', 'torsion': [48]}
id: 3860415
{'degree': 16, 'field': [156163135476069804096, -2200725616876351488, 15506839133356032, 67296410242022976, 170104925316592, 56290973654048, 13690072447520, 293920824064, 3176361084, 2738503696, 82986640, -1597228, 261477, 12674, 2, -2, 1], 'lmfdb_label': '82110.bs6', 'torsion': [4, 16]}
id: 3860416
{'degree': 16, 'field': [962108000389909872640, 989424388102225920, -976346068868339200, -26169397681624960, 13116575730034976, -38193090253032, 12632493130460, 84277797690, -20840689649, -153599670, 31823146, -614570, 113136, -4970, 730, -8, 1], 'lmfdb_label': '82110.bs6', 'torsion': [2, 32]}
id: 3860417
{'degree': 16, 'field': [1737472494335424004096, 0, -113312268579715046400, 0, 2023242359452276096, 0, -13997286861877500, 0, 38439759693996, 0, -49446095235, 0, 31188871, 0, -9165, 0, 1], 'lmfdb_label': '82110.bs6', 'torsion': [2, 32]}
id: 3860419
{'degree': 16, 'field': [176929112778756448022784, 0, -266017601971495930368, 0, -1726580059384963040, 0, -1891724255984256, 0, 4201370442885, 0, 1054055356, 0, -4499922, 0, 0, 0, 1], 'lmfdb_label': '82110.bs6', 'torsion': [48]}

Column	Type
conductor	integer
conductor_valuation	smallint
discriminant_valuation	smallint
j_denominator_valuation	smallint
kodaira_symbol	smallint
lmfdb_label	text
prime	integer
reduction_type	smallint
root_number	smallint
tamagawa_number	smallint

ec_localdata • Show schema

id: 2000537
{'conductor': 82110, 'conductor_valuation': 1, 'discriminant_valuation': 32, 'j_denominator_valuation': 32, 'kodaira_symbol': 36, 'lmfdb_label': '82110.bs6', 'prime': 2, 'reduction_type': 1, 'root_number': -1, 'tamagawa_number': 32}
id: 2000538
{'conductor': 82110, 'conductor_valuation': 1, 'discriminant_valuation': 4, 'j_denominator_valuation': 4, 'kodaira_symbol': 8, 'lmfdb_label': '82110.bs6', 'prime': 3, 'reduction_type': 1, 'root_number': -1, 'tamagawa_number': 4}
id: 2000539
{'conductor': 82110, 'conductor_valuation': 1, 'discriminant_valuation': 4, 'j_denominator_valuation': 4, 'kodaira_symbol': 8, 'lmfdb_label': '82110.bs6', 'prime': 5, 'reduction_type': 1, 'root_number': -1, 'tamagawa_number': 4}
id: 2000540
{'conductor': 82110, 'conductor_valuation': 1, 'discriminant_valuation': 1, 'j_denominator_valuation': 1, 'kodaira_symbol': 5, 'lmfdb_label': '82110.bs6', 'prime': 7, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 1}
id: 2000541
{'conductor': 82110, 'conductor_valuation': 1, 'discriminant_valuation': 2, 'j_denominator_valuation': 2, 'kodaira_symbol': 6, 'lmfdb_label': '82110.bs6', 'prime': 17, 'reduction_type': 1, 'root_number': -1, 'tamagawa_number': 2}
id: 2000542
{'conductor': 82110, 'conductor_valuation': 1, 'discriminant_valuation': 1, 'j_denominator_valuation': 1, 'kodaira_symbol': 5, 'lmfdb_label': '82110.bs6', 'prime': 23, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 1}

Column	Type
adelic_gens	bigint[]
adelic_image	text
conductor	integer
elladic_image	text
lmfdb_label	text
modell_image	text
prime	smallint

ec_galrep • Show schema

id: 2205382
{'adelic_gens': [[1, 0, 32, 1], [1051031, 26, 260214, 1310891], [1, 32, 0, 1], [850103, 26, 619014, 875], [5, 28, 68, 381], [492663, 2, 492634, 1313743], [23, 18, 1311198, 1311755], [938432, 29, 753547, 2562], [742590, 31, 685289, 1313604], [1313729, 32, 1313728, 33], [875863, 26, 1311222, 1310891], [1231681, 32, 247290, 991]], 'adelic_image': '1313760.768.13.?', 'conductor': 82110, 'lmfdb_label': '82110.bs6', 'prime': 0}
id: 336420
{'conductor': 82110, 'elladic_image': '16.96.0.95', 'lmfdb_label': '82110.bs6', 'modell_image': '2B', 'prime': 2}

Column	Type
lmfdb_iso	text
p	smallint
prec	smallint
unit	numeric
val	smallint

ec_padic • Show schema

id: 4236030
{'lmfdb_iso': '82110.bs', 'p': 11, 'prec': 20, 'unit': 32702201256068778573, 'val': 1}
id: 4236028
{'lmfdb_iso': '82110.bs', 'p': 13, 'prec': 20, 'unit': 170422566261904553613, 'val': 1}
id: 4236031
{'lmfdb_iso': '82110.bs', 'p': 19, 'prec': 20, 'unit': 626463097538166343515799, 'val': 1}
id: 4236029
{'lmfdb_iso': '82110.bs', 'p': 29, 'prec': 20, 'unit': 1274550482482936693442952409, 'val': 1}
id: 4236024
{'lmfdb_iso': '82110.bs', 'p': 37, 'prec': 20, 'unit': 153709385256265349528642532492, 'val': 1}
id: 4236026
{'lmfdb_iso': '82110.bs', 'p': 41, 'prec': 20, 'unit': 4124505813912686061724912329002, 'val': 1}
id: 4236027
{'lmfdb_iso': '82110.bs', 'p': 43, 'prec': 20, 'unit': 313794116651113313610543150545, 'val': 1}
id: 4236032
{'lmfdb_iso': '82110.bs', 'p': 53, 'prec': 20, 'unit': 514079795423136305242908110955302, 'val': 1}
id: 4236035
{'lmfdb_iso': '82110.bs', 'p': 59, 'prec': 20, 'unit': 1373180473342419009630894594773602, 'val': 1}
id: 4236036
{'lmfdb_iso': '82110.bs', 'p': 61, 'prec': 20, 'unit': 7849164803061490415256332888441995, 'val': 1}
id: 4236023
{'lmfdb_iso': '82110.bs', 'p': 67, 'prec': 20, 'unit': 20277641553557277290824445517887470, 'val': 1}
id: 4236025
{'lmfdb_iso': '82110.bs', 'p': 71, 'prec': 20, 'unit': 73271211922403488414681125325869296, 'val': 1}
id: 4236033
{'lmfdb_iso': '82110.bs', 'p': 73, 'prec': 20, 'unit': 185266699369358011617986951364440946, 'val': 1}
id: 4236022
{'lmfdb_iso': '82110.bs', 'p': 83, 'prec': 20, 'unit': 2576603744706529038439982148985526394, 'val': 1}
id: 4236034
{'lmfdb_iso': '82110.bs', 'p': 89, 'prec': 20, 'unit': 3214872213493159338462367919601441120, 'val': 1}
id: 4236021
{'lmfdb_iso': '82110.bs', 'p': 97, 'prec': 20, 'unit': 55070366777344394997085987446012228866, 'val': 1}

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.