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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 412698bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 412698.bg4 | 412698bg1 | \([1, 0, 1, -259926, 18688360]\) | \(402355893390625/201513996288\) | \(972669570908884992\) | \([2]\) | \(7464960\) | \(2.1443\) | \(\Gamma_0(N)\)-optimal |
| 412698.bg3 | 412698bg2 | \([1, 0, 1, -2260886, -1295542168]\) | \(264788619837198625/3058196150592\) | \(14761328703442820928\) | \([2]\) | \(14929920\) | \(2.4909\) | |
| 412698.bg2 | 412698bg3 | \([1, 0, 1, -11413926, -14842722776]\) | \(34069730739753390625/1354703543952\) | \(6538895258279409168\) | \([2]\) | \(22394880\) | \(2.6937\) | |
| 412698.bg1 | 412698bg4 | \([1, 0, 1, -182621066, -949907638600]\) | \(139545621883503188502625/220644468\) | \(1065008703942612\) | \([2]\) | \(44789760\) | \(3.0402\) |
Rank
sage: E.rank()
The elliptic curves in class 412698bg have rank \(0\).
Complex multiplication
The elliptic curves in class 412698bg do not have complex multiplication.Modular form 412698.2.a.bg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
