Properties

Label 19110j
Number of curves 88
Conductor 1911019110
CM no
Rank 11
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 19110j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19110.m7 19110j1 [1,1,0,167752,39126464][1, 1, 0, -167752, -39126464] 4437543642183289/3033210136320-4437543642183289/3033210136320 356854139327911680-356854139327911680 [2][2] 331776331776 2.06842.0684 Γ0(N)\Gamma_0(N)-optimal
19110.m6 19110j2 [1,1,0,3025432,2026357136][1, 1, 0, -3025432, -2026357136] 26031421522845051769/579778977960026031421522845051769/5797789779600 682104169780160400682104169780160400 [2,2][2, 2] 663552663552 2.41502.4150  
19110.m8 19110j3 [1,1,0,1360313,582407029][1, 1, 0, 1360313, 582407029] 2366200373628880151/26124201492480002366200373628880151/2612420149248000 307348618138877952000-307348618138877952000 [2][2] 995328995328 2.61772.6177  
19110.m3 19110j4 [1,1,0,48404332,129640899716][1, 1, 0, -48404332, -129640899716] 106607603143751752938169/5290068420106607603143751752938169/5290068420 622371259544580622371259544580 [2][2] 13271041327104 2.76162.7616  
19110.m5 19110j5 [1,1,0,3369412,1537423964][1, 1, 0, -3369412, -1537423964] 35958207000163259449/1214572951887750035958207000163259449/12145729518877500 14289329321664189975001428932932166418997500 [2][2] 13271041327104 2.76162.7616  
19110.m4 19110j6 [1,1,0,7671367,5454095221][1, 1, 0, -7671367, 5454095221] 424378956393532177129/136231857216000000424378956393532177129/136231857216000000 1602754176960518400000016027541769605184000000 [2,2][2, 2] 19906561990656 2.96432.9643  
19110.m2 19110j7 [1,1,0,48831367,127237512779][1, 1, 0, -48831367, -127237512779] 109454124781830273937129/3914078300576808000109454124781830273937129/3914078300576808000 460487397984560884392000460487397984560884392000 [2][2] 39813123981312 3.31093.3109  
19110.m1 19110j8 [1,1,0,111018247,450114381109][1, 1, 0, -111018247, 450114381109] 1286229821345376481036009/2472654843750000001286229821345376481036009/247265484375000000 2909053697123437500000029090536971234375000000 [2][2] 39813123981312 3.31093.3109  

Rank

sage: E.rank()
 

The elliptic curves in class 19110j have rank 11.

Complex multiplication

The elliptic curves in class 19110j do not have complex multiplication.

Modular form 19110.2.a.j

sage: E.q_eigenform(10)
 
qq2q3+q4+q5+q6q8+q9q10q12q13q15+q16+6q17q18+4q19+O(q20)q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} - q^{13} - q^{15} + q^{16} + 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The i,ji,j entry is the smallest degree of a cyclic isogeny between the ii-th and jj-th curve in the isogeny class, in the Cremona numbering.

(1234461212216223663611212244421214631242124161236326612212643122141264123241)\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.