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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 96330bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 96330.bg4 | 96330bc1 | \([1, 0, 1, 1577611, -1621608064]\) | \(89962967236397039/287450726400000\) | \(-1387469753244057600000\) | \([2]\) | \(5184000\) | \(2.7400\) | \(\Gamma_0(N)\)-optimal |
| 96330.bg3 | 96330bc2 | \([1, 0, 1, -14862709, -19002314368]\) | \(75224183150104868881/11219310000000000\) | \(54153466481790000000000\) | \([2]\) | \(10368000\) | \(3.0866\) | |
| 96330.bg2 | 96330bc3 | \([1, 0, 1, -557947589, -5072736926944]\) | \(-3979640234041473454886161/1471455901872240\) | \(-7102436590260044882160\) | \([2]\) | \(25920000\) | \(3.5447\) | |
| 96330.bg1 | 96330bc4 | \([1, 0, 1, -8927162209, -324652871034568]\) | \(16300610738133468173382620881/2228489100\) | \(10756491244281900\) | \([2]\) | \(51840000\) | \(3.8913\) |
Rank
sage: E.rank()
The elliptic curves in class 96330bc have rank \(1\).
Complex multiplication
The elliptic curves in class 96330bc do not have complex multiplication.Modular form 96330.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
