Properties

Label 2415.i
Number of curves $2$
Conductor $2415$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2415.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2415.i1 2415g1 \([1, 0, 1, -1438, -21037]\) \(328523283207001/1109390625\) \(1109390625\) \([2]\) \(1152\) \(0.59964\) \(\Gamma_0(N)\)-optimal
2415.i2 2415g2 \([1, 0, 1, -813, -39287]\) \(-59323563117001/630142750125\) \(-630142750125\) \([2]\) \(2304\) \(0.94621\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 2415.2.a.i

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - q^{7} - 3 q^{8} + q^{9} + q^{10} - 2 q^{11} - q^{12} - q^{14} + q^{15} - q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.