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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 493680be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
493680.be3 | 493680be1 | \([0, -1, 0, -268176, -44735040]\) | \(293946977449/50490000\) | \(366371286589440000\) | \([2]\) | \(6635520\) | \(2.0906\) | \(\Gamma_0(N)\)-optimal* |
493680.be2 | 493680be2 | \([0, -1, 0, -1236176, 487277760]\) | \(28790481449449/2549240100\) | \(18498086259900825600\) | \([2, 2]\) | \(13271040\) | \(2.4371\) | \(\Gamma_0(N)\)-optimal* |
493680.be1 | 493680be3 | \([0, -1, 0, -19337776, 32737088320]\) | \(110211585818155849/993794670\) | \(7211286033939947520\) | \([2]\) | \(26542080\) | \(2.7837\) | \(\Gamma_0(N)\)-optimal* |
493680.be4 | 493680be4 | \([0, -1, 0, 1377424, 2270798400]\) | \(39829997144951/330164359470\) | \(-2395776216379525816320\) | \([2]\) | \(26542080\) | \(2.7837\) |
Rank
sage: E.rank()
The elliptic curves in class 493680be have rank \(0\).
Complex multiplication
The elliptic curves in class 493680be do not have complex multiplication.Modular form 493680.2.a.be
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.