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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 42630dl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42630.dh2 | 42630dl1 | \([1, 0, 0, 7790, -325018]\) | \(1066931459038991/1552488867810\) | \(-76071954522690\) | \([]\) | \(178752\) | \(1.3491\) | \(\Gamma_0(N)\)-optimal |
42630.dh1 | 42630dl2 | \([1, 0, 0, -2631210, 1642616100]\) | \(-41114420704407863185009/1387061010000000\) | \(-67965989490000000\) | \([7]\) | \(1251264\) | \(2.3220\) |
Rank
sage: E.rank()
The elliptic curves in class 42630dl have rank \(0\).
Complex multiplication
The elliptic curves in class 42630dl do not have complex multiplication.Modular form 42630.2.a.dl
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.