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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 485100cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.cm2 | 485100cm1 | \([0, 0, 0, 88200, -20794375]\) | \(95551488/290521\) | \(-230711659620750000\) | \([2]\) | \(3686400\) | \(2.0150\) | \(\Gamma_0(N)\)-optimal* |
485100.cm1 | 485100cm2 | \([0, 0, 0, -812175, -241386250]\) | \(4662947952/717409\) | \(9115464755628000000\) | \([2]\) | \(7372800\) | \(2.3616\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485100cm have rank \(1\).
Complex multiplication
The elliptic curves in class 485100cm do not have complex multiplication.Modular form 485100.2.a.cm
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.