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Properties

Label 462462.x
Number of curves $2$
Conductor $462462$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("x1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 462462.x have rank \(0\).

Complex multiplication

The elliptic curves in class 462462.x do not have complex multiplication.

Modular form 462462.2.a.x

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - q^{12} - q^{13} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 462462.x

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462462.x1 462462x2 \([1, 1, 0, -429514670, 2100752080596]\) \(31588958858820875/11407036691232\) \(3164428191879285158951611488\) \([2]\) \(259522560\) \(3.9775\) \(\Gamma_0(N)\)-optimal*
462462.x2 462462x1 \([1, 1, 0, 81802290, 231479538228]\) \(218221573115125/209703416832\) \(-58173864266313989767461888\) \([2]\) \(129761280\) \(3.6309\) \(\Gamma_0(N)\)-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 462462.x1.