Properties

Label 690f
Number of curves $4$
Conductor $690$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("690.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 690f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
690.f3 690f1 [1, 0, 1, -13, 8] [2] 64 \(\Gamma_0(N)\)-optimal
690.f2 690f2 [1, 0, 1, -93, -344] [2, 2] 128  
690.f1 690f3 [1, 0, 1, -1473, -21872] [2] 256  
690.f4 690f4 [1, 0, 1, 7, -1024] [4] 256  

Rank

sage: E.rank()
 

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

Modular form 690.2.a.f

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

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.