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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 489762dm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.dm2 | 489762dm1 | \([1, -1, 1, -6116, -167709]\) | \(7189057/644\) | \(2266070982084\) | \([2]\) | \(1244160\) | \(1.1105\) | \(\Gamma_0(N)\)-optimal* |
489762.dm1 | 489762dm2 | \([1, -1, 1, -21326, 1012587]\) | \(304821217/51842\) | \(182418714057762\) | \([2]\) | \(2488320\) | \(1.4571\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 489762dm have rank \(1\).
Complex multiplication
The elliptic curves in class 489762dm do not have complex multiplication.Modular form 489762.2.a.dm
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.