Properties

Label 489762.a
Number of curves $2$
Conductor $489762$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 489762.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
489762.a1 489762a2 \([1, -1, 0, -110304, 3578134]\) \(42180533641/22862322\) \(80446652899473042\) \([2]\) \(7077888\) \(1.9348\) \(\Gamma_0(N)\)-optimal*
489762.a2 489762a1 \([1, -1, 0, 26586, 429664]\) \(590589719/365148\) \(-1284862246841628\) \([2]\) \(3538944\) \(1.5882\) \(\Gamma_0(N)\)-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 489762.a1.

Rank

sage: E.rank()
 

The elliptic curves in class 489762.a have rank \(2\).

Complex multiplication

The elliptic curves in class 489762.a do not have complex multiplication.

Modular form 489762.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 4 q^{5} - q^{7} - q^{8} + 4 q^{10} + 2 q^{11} + q^{14} + q^{16} - 2 q^{17} + 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.