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Properties

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

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

Elliptic curves in class 424830f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
424830.f1 424830f1 [1, 1, 0, -470124253, -4030659158147] [] 274827168 \(\Gamma_0(N)\)-optimal*
424830.f2 424830f2 [1, 1, 0, 2184001172, -14826048911792] [] 824481504 \(\Gamma_0(N)\)-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 424830f1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 424830.2.a.f

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

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 Cremona 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 Cremona labels.