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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 323400bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 323400.bi3 | 323400bi1 | \([0, -1, 0, -4825221508, -129006433180988]\) | \(26401417552259125806544/507547744790625\) | \(238849938507488962500000000\) | \([2]\) | \(283115520\) | \(4.1845\) | \(\Gamma_0(N)\)-optimal |
| 323400.bi2 | 323400bi2 | \([0, -1, 0, -4985966008, -119951052517988]\) | \(7282213870869695463556/912102595400390625\) | \(1716927331940168906250000000000\) | \([2, 2]\) | \(566231040\) | \(4.5310\) | |
| 323400.bi1 | 323400bi3 | \([0, -1, 0, -19961003008, 961396369252012]\) | \(233632133015204766393938/29145526885986328125\) | \(109726146963500976562500000000000\) | \([2]\) | \(1132462080\) | \(4.8776\) | |
| 323400.bi4 | 323400bi4 | \([0, -1, 0, 7417158992, -621756683767988]\) | \(11986661998777424518222/51295853620928503125\) | \(-193116988244755758852900000000000\) | \([2]\) | \(1132462080\) | \(4.8776\) |
Rank
sage: E.rank()
The elliptic curves in class 323400bi have rank \(1\).
Complex multiplication
The elliptic curves in class 323400bi do not have complex multiplication.Modular form 323400.2.a.bi
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.
