Properties

Label 2415.a
Number of curves 4
Conductor 2415
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("2415.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2415.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2415.a1 2415a3 [1, 1, 1, -5051, 136064] [2] 1920  
2415.a2 2415a2 [1, 1, 1, -326, 1874] [2, 2] 960  
2415.a3 2415a1 [1, 1, 1, -81, -282] [2] 480 \(\Gamma_0(N)\)-optimal
2415.a4 2415a4 [1, 1, 1, 479, 10568] [2] 1920  

Rank

sage: E.rank()
 

The elliptic curves in class 2415.a have rank \(1\).

Modular form 2415.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - q^{7} + 3q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} - 6q^{13} + q^{14} + q^{15} - q^{16} - 2q^{17} - q^{18} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.