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SageMath
E = EllipticCurve("bs1")
E.isogeny_class()
Elliptic curves in class 406560bs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
406560.bs3 | 406560bs1 | \([0, -1, 0, -8510, 300192]\) | \(601211584/11025\) | \(1250013441600\) | \([2, 2]\) | \(983040\) | \(1.1160\) | \(\Gamma_0(N)\)-optimal* |
406560.bs1 | 406560bs2 | \([0, -1, 0, -135560, 19256052]\) | \(303735479048/105\) | \(95239119360\) | \([2]\) | \(1966080\) | \(1.4625\) | \(\Gamma_0(N)\)-optimal* |
406560.bs4 | 406560bs3 | \([0, -1, 0, -40, 862600]\) | \(-8/354375\) | \(-321432027840000\) | \([2]\) | \(1966080\) | \(1.4625\) | |
406560.bs2 | 406560bs4 | \([0, -1, 0, -17585, -442143]\) | \(82881856/36015\) | \(261336143523840\) | \([2]\) | \(1966080\) | \(1.4625\) |
Rank
sage: E.rank()
The elliptic curves in class 406560bs have rank \(1\).
Complex multiplication
The elliptic curves in class 406560bs do not have complex multiplication.Modular form 406560.2.a.bs
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.