Properties

Label 690.g
Number of curves $2$
Conductor $690$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("690.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 690.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
690.g1 690h2 [1, 1, 1, -116, 413] [2] 192  
690.g2 690h1 [1, 1, 1, 4, 29] [2] 96 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 690.g have rank \(1\).

Modular form 690.2.a.g

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

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.