Properties

Label 317680.s
Number of curves 4
Conductor 317680
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("317680.s1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 317680.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
317680.s1 317680s4 [0, 1, 0, -2563220, -1580381000] [2] 5971968  
317680.s2 317680s3 [0, 1, 0, -160765, -24551142] [2] 2985984  
317680.s3 317680s2 [0, 1, 0, -36220, -1511400] [2] 1990656  
317680.s4 317680s1 [0, 1, 0, -16365, 783838] [2] 995328 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 317680.s have rank \(1\).

Modular form 317680.2.a.s

sage: E.q_eigenform(10)
 
\( q - 2q^{3} + q^{5} + 4q^{7} + q^{9} + q^{11} + 4q^{13} - 2q^{15} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.