Properties

Label 92697e
Number of curves 4
Conductor 92697
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("92697.c1")
sage: E.isogeny_class()

Elliptic curves in class 92697e

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
92697.c3 92697e1 [1, 0, 0, -18317, -947688] 2 224640 \(\Gamma_0(N)\)-optimal
92697.c2 92697e2 [1, 0, 0, -32362, 701195] 4 449280  
92697.c4 92697e3 [1, 0, 0, 122133, 5490540] 2 898560  
92697.c1 92697e4 [1, 0, 0, -411577, 101496542] 2 898560  

Rank

sage: E.rank()

The elliptic curves in class 92697e have rank \(1\).

Modular form 92697.2.a.c

sage: E.q_eigenform(10)
\( q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} + 4q^{7} + 3q^{8} + q^{9} - 2q^{10} + q^{11} - q^{12} - 2q^{13} - 4q^{14} + 2q^{15} - q^{16} - 2q^{17} - q^{18} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.