Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2415.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2415.d1 2415i1 \([1, 0, 0, -230, -1173]\) \(1345938541921/203765625\) \(203765625\) \([2]\) \(768\) \(0.31835\) \(\Gamma_0(N)\)-optimal
2415.d2 2415i2 \([1, 0, 0, 395, -6298]\) \(6814692748079/21258460125\) \(-21258460125\) \([2]\) \(1536\) \(0.66492\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 2415.2.a.d

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