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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1690.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1690.g1 | 1690h1 | \([1, 0, 0, -10, 12]\) | \(-658489/40\) | \(-6760\) | \([]\) | \(144\) | \(-0.51021\) | \(\Gamma_0(N)\)-optimal |
1690.g2 | 1690h2 | \([1, 0, 0, 55, 25]\) | \(108750551/64000\) | \(-10816000\) | \([]\) | \(432\) | \(0.039092\) |
Rank
sage: E.rank()
The elliptic curves in class 1690.g have rank \(1\).
Complex multiplication
The elliptic curves in class 1690.g do not have complex multiplication.Modular form 1690.2.a.g
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.