Properties

Label 690b
Number of curves $2$
Conductor $690$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("690.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 690b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
690.b2 690b1 [1, 1, 0, 167, -347] [2] 336 \(\Gamma_0(N)\)-optimal
690.b1 690b2 [1, 1, 0, -753, -3843] [2] 672  

Rank

sage: E.rank()
 

The elliptic curves in class 690b have rank \(0\).

Modular form 690.2.a.b

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

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 Cremona 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 Cremona labels.