Properties

Label 493680.cp
Number of curves 2
Conductor 493680
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("493680.cp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 493680.cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
493680.cp1 493680cp2 [0, -1, 0, -22767400, 41797155952] [2] 30965760 \(\Gamma_0(N)\)-optimal*
493680.cp2 493680cp1 [0, -1, 0, -1703720, 377535600] [2] 15482880 \(\Gamma_0(N)\)-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 493680.cp2.

Rank

sage: E.rank()
 

The elliptic curves in class 493680.cp have rank \(0\).

Modular form 493680.2.a.cp

sage: E.q_eigenform(10)
 
\( q - q^{3} + q^{5} - 2q^{7} + q^{9} + 4q^{13} - q^{15} - q^{17} - 6q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.