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SageMath
E = EllipticCurve("pr1")
E.isogeny_class()
Elliptic curves in class 331200.pr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
331200.pr1 | 331200pr2 | \([0, 0, 0, -3492300, -2510822000]\) | \(1577505447721/838350\) | \(2503299686400000000\) | \([2]\) | \(10616832\) | \(2.4801\) | |
331200.pr2 | 331200pr1 | \([0, 0, 0, -180300, -53318000]\) | \(-217081801/285660\) | \(-852976189440000000\) | \([2]\) | \(5308416\) | \(2.1335\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 331200.pr have rank \(0\).
Complex multiplication
The elliptic curves in class 331200.pr do not have complex multiplication.Modular form 331200.2.a.pr
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.