Properties

Label 337896f
Number of curves 44
Conductor 337896337896
CM no
Rank 22
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 337896f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
337896.f3 337896f1 [0,0,0,23826,816221][0, 0, 0, -23826, 816221] 2725888/10532725888/1053 577826543251152577826543251152 [2][2] 884736884736 1.53151.5315 Γ0(N)\Gamma_0(N)-optimal
337896.f2 337896f2 [0,0,0,170031,26407150][0, 0, 0, -170031, -26407150] 61918288/152161918288/1521 1335421344402662413354213444026624 [2,2][2, 2] 17694721769472 1.87811.8781  
337896.f4 337896f3 [0,0,0,24909,83446594][0, 0, 0, 24909, -83446594] 48668/8568348668/85683 3009149429387332608-3009149429387332608 [2][2] 35389443538944 2.22472.2247  
337896.f1 337896f4 [0,0,0,2704251,1711663450][0, 0, 0, -2704251, -1711663450] 62275269892/3962275269892/39 13696629173360641369662917336064 [2][2] 35389443538944 2.22472.2247  

Rank

sage: E.rank()
 

The elliptic curves in class 337896f have rank 22.

Complex multiplication

The elliptic curves in class 337896f do not have complex multiplication.

Modular form 337896.2.a.f

sage: E.q_eigenform(10)
 
q2q5q132q17+O(q20)q - 2 q^{5} - q^{13} - 2 q^{17} + 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.

(1244212242144241)\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with Cremona labels.