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Properties

Label 462462a
Number of curves $2$
Conductor $462462$
CM no
Rank $1$
Graph

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 462462a have rank \(1\).

Complex multiplication

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

Modular form 462462.2.a.a

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 4 q^{5} + q^{6} - q^{8} + q^{9} + 4 q^{10} - q^{12} - q^{13} + 4 q^{15} + q^{16} + 4 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.

Elliptic curves in class 462462a

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462462.a2 462462a1 \([1, 1, 0, -6052, -3159680]\) \(-117649/20592\) \(-4291833650792688\) \([2]\) \(5529600\) \(1.6790\) \(\Gamma_0(N)\)-optimal*
462462.a1 462462a2 \([1, 1, 0, -361792, -83201180]\) \(25128011089/245388\) \(51144351005279532\) \([2]\) \(11059200\) \(2.0255\) \(\Gamma_0(N)\)-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 462462a1.