Properties

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

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

Elliptic curves in class 2415.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2415.a1 2415a3 \([1, 1, 1, -5051, 136064]\) \(14251520160844849/264449745\) \(264449745\) \([2]\) \(1920\) \(0.74103\)  
2415.a2 2415a2 \([1, 1, 1, -326, 1874]\) \(3832302404449/472410225\) \(472410225\) \([2, 2]\) \(960\) \(0.39445\)  
2415.a3 2415a1 \([1, 1, 1, -81, -282]\) \(58818484369/7455105\) \(7455105\) \([2]\) \(480\) \(0.047881\) \(\Gamma_0(N)\)-optimal
2415.a4 2415a4 \([1, 1, 1, 479, 10568]\) \(12152722588271/53476250625\) \(-53476250625\) \([2]\) \(1920\) \(0.74103\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 2415.a do not have complex multiplication.

Modular form 2415.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.