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Properties

Label 2499.d
Number of curves $2$
Conductor $2499$
CM no
Rank $1$
Graph

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

Elliptic curves in class 2499.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2499.d1 2499d2 \([0, -1, 1, -2907, 61340]\) \(-23100424192/14739\) \(-1734028611\) \([]\) \(1728\) \(0.71373\)  
2499.d2 2499d1 \([0, -1, 1, 33, 335]\) \(32768/459\) \(-54000891\) \([]\) \(576\) \(0.16443\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 2499.d have rank \(1\).

Complex multiplication

The elliptic curves in class 2499.d do not have complex multiplication.

Modular form 2499.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{4} - 3 q^{5} + q^{9} - 3 q^{11} + 2 q^{12} + q^{13} + 3 q^{15} + 4 q^{16} + q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.