Properties

Label 424830.ej
Number of curves $2$
Conductor $424830$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ej1")
 
E.isogeny_class()
 

Elliptic curves in class 424830.ej

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424830.ej1 424830ej2 \([1, 0, 1, -221958798, 1272771222328]\) \(-12242088317612041/600\) \(-59270220673200600\) \([]\) \(49968576\) \(3.1421\) \(\Gamma_0(N)\)-optimal*
424830.ej2 424830ej1 \([1, 0, 1, -2716173, 1777876678]\) \(-22434194041/843750\) \(-83348747821688343750\) \([]\) \(16656192\) \(2.5928\) \(\Gamma_0(N)\)-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 424830.ej1.

Rank

sage: E.rank()
 

The elliptic curves in class 424830.ej have rank \(0\).

Complex multiplication

The elliptic curves in class 424830.ej do not have complex multiplication.

Modular form 424830.2.a.ej

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} + 3 q^{11} + q^{12} - 2 q^{13} + q^{15} + q^{16} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.