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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 64350bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 64350.f4 | 64350bo1 | \([1, -1, 0, 4833, 7359741]\) | \(1095912791/2055596400\) | \(-23414527743750000\) | \([2]\) | \(589824\) | \(1.8200\) | \(\Gamma_0(N)\)-optimal |
| 64350.f3 | 64350bo2 | \([1, -1, 0, -539667, 149474241]\) | \(1525998818291689/37268302500\) | \(424509258164062500\) | \([2, 2]\) | \(1179648\) | \(2.1665\) | |
| 64350.f2 | 64350bo3 | \([1, -1, 0, -1207917, -293575509]\) | \(17111482619973769/6627044531250\) | \(75486179113769531250\) | \([2]\) | \(2359296\) | \(2.5131\) | |
| 64350.f1 | 64350bo4 | \([1, -1, 0, -8583417, 9681317991]\) | \(6139836723518159689/3799803150\) | \(43282132755468750\) | \([2]\) | \(2359296\) | \(2.5131\) |
Rank
sage: E.rank()
The elliptic curves in class 64350bo have rank \(1\).
Complex multiplication
The elliptic curves in class 64350bo do not have complex multiplication.Modular form 64350.2.a.bo
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.
