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Properties

Label 6762.d
Number of curves $2$
Conductor $6762$
CM no
Rank $1$
Graph

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

Elliptic curves in class 6762.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6762.d1 6762g1 \([1, 1, 0, -782261, 265976985]\) \(1311889499494111/438012\) \(17675364109284\) \([2]\) \(53760\) \(1.8999\) \(\Gamma_0(N)\)-optimal
6762.d2 6762g2 \([1, 1, 0, -778831, 268429435]\) \(-1294708239486271/23981814018\) \(-967752698029462926\) \([2]\) \(107520\) \(2.2465\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6762.d have rank \(1\).

Complex multiplication

The elliptic curves in class 6762.d do not have complex multiplication.

Modular form 6762.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} - q^{8} + q^{9} + 2 q^{10} + 2 q^{11} - q^{12} - 2 q^{13} + 2 q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.