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SageMath
sage: E = EllipticCurve("e1")
sage: E.isogeny_class()
Elliptic curves in class 1989.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 1989.e1 | 1989e5 | [1, -1, 0, -181566, 29819155] | [2] | 8192 | |
| 1989.e2 | 1989e3 | [1, -1, 0, -12501, 368032] | [2, 2] | 4096 | |
| 1989.e3 | 1989e2 | [1, -1, 0, -4896, -126293] | [2, 2] | 2048 | |
| 1989.e4 | 1989e1 | [1, -1, 0, -4851, -128840] | [2] | 1024 | \(\Gamma_0(N)\)-optimal |
| 1989.e5 | 1989e4 | [1, -1, 0, 1989, -458150] | [2] | 4096 | |
| 1989.e6 | 1989e6 | [1, -1, 0, 34884, 2462449] | [2] | 8192 |
Rank
sage: E.rank()
The elliptic curves in class 1989.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1989.e do not have complex multiplication.Modular form 1989.2.a.e
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
