Properties

Label 493680eg
Number of curves $2$
Conductor $493680$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("eg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 493680eg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
493680.eg2 493680eg1 [0, 1, 0, 51264, 3926580] [2] 2949120 \(\Gamma_0(N)\)-optimal*
493680.eg1 493680eg2 [0, 1, 0, -277856, 35917044] [2] 5898240 \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 493680eg1.

Rank

sage: E.rank()
 

The elliptic curves in class 493680eg have rank \(2\).

Complex multiplication

The elliptic curves in class 493680eg do not have complex multiplication.

Modular form 493680.2.a.eg

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.