Properties

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

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

Elliptic curves in class 690g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.h4 690g1 \([1, 1, 1, -4491, -207687]\) \(-10017490085065009/12502381363200\) \(-12502381363200\) \([4]\) \(1792\) \(1.2065\) \(\Gamma_0(N)\)-optimal
690.h3 690g2 \([1, 1, 1, -86411, -9808711]\) \(71356102305927901489/35540674560000\) \(35540674560000\) \([2, 2]\) \(3584\) \(1.5531\)  
690.h1 690g3 \([1, 1, 1, -1382411, -626186311]\) \(292169767125103365085489/72534787200\) \(72534787200\) \([2]\) \(7168\) \(1.8996\)  
690.h2 690g4 \([1, 1, 1, -101131, -6258247]\) \(114387056741228939569/49503729150000000\) \(49503729150000000\) \([2]\) \(7168\) \(1.8996\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 690g do not have complex multiplication.

Modular form 690.2.a.g

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