Properties

Label 457776er
Number of curves $2$
Conductor $457776$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("er1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 457776er

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
457776.er2 457776er1 [0, 0, 0, -500259, 110375458] [2] 5308416 \(\Gamma_0(N)\)-optimal*
457776.er1 457776er2 [0, 0, 0, -7574979, 8024157250] [2] 10616832 \(\Gamma_0(N)\)-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 457776er1.

Rank

sage: E.rank()
 

The elliptic curves in class 457776er have rank \(0\).

Complex multiplication

The elliptic curves in class 457776er do not have complex multiplication.

Modular form 457776.2.a.er

sage: E.q_eigenform(10)
 
\( q + 2q^{5} - 2q^{7} + q^{11} + 4q^{13} - 2q^{19} + O(q^{20}) \)

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 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.