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SageMath
sage: E = EllipticCurve("3136.e1")
sage: E.isogeny_class()
Elliptic curves in class 3136k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 3136.e5 | 3136k1 | [0, 1, 0, -1633, -51969] | [2] | 3072 | \(\Gamma_0(N)\)-optimal |
| 3136.e4 | 3136k2 | [0, 1, 0, -32993, -2316161] | [2] | 6144 | |
| 3136.e6 | 3136k3 | [0, 1, 0, 14047, 1080127] | [2] | 9216 | |
| 3136.e3 | 3136k4 | [0, 1, 0, -111393, 11692351] | [2] | 18432 | |
| 3136.e2 | 3136k5 | [0, 1, 0, -534753, 150770815] | [2] | 27648 | |
| 3136.e1 | 3136k6 | [0, 1, 0, -8562913, 9641661567] | [2] | 55296 |
Rank
sage: E.rank()
The elliptic curves in class 3136k have rank \(0\).
Modular form 3136.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
