Properties

Label 424830.f
Number of curves $2$
Conductor $424830$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 424830.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424830.f1 424830f1 \([1, 1, 0, -470124253, -4030659158147]\) \(-5831629329001/186624000\) \(-367738639324223161026816000\) \([]\) \(274827168\) \(3.8733\) \(\Gamma_0(N)\)-optimal*
424830.f2 424830f2 \([1, 1, 0, 2184001172, -14826048911792]\) \(584669638645799/386547056640\) \(-761682788085010617870330101760\) \([]\) \(824481504\) \(4.4226\) \(\Gamma_0(N)\)-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.f1.

Rank

sage: E.rank()
 

The elliptic curves in class 424830.f have rank \(1\).

Complex multiplication

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

Modular form 424830.2.a.f

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