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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 493680o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
493680.o1 | 493680o1 | \([0, -1, 0, -31016, -2586384]\) | \(-55025549689/16730550\) | \(-1003323320524800\) | \([]\) | \(2115072\) | \(1.5924\) | \(\Gamma_0(N)\)-optimal* |
493680.o2 | 493680o2 | \([0, -1, 0, 230344, 22086000]\) | \(22538336092151/16581375000\) | \(-994377364992000000\) | \([]\) | \(6345216\) | \(2.1417\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 493680o have rank \(1\).
Complex multiplication
The elliptic curves in class 493680o do not have complex multiplication.Modular form 493680.2.a.o
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.