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SageMath
sage: E = EllipticCurve("6720.b1")
sage: E.isogeny_class()
Elliptic curves in class 6720e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 6720.b5 | 6720e1 | [0, -1, 0, -61, 781] | [2] | 2048 | \(\Gamma_0(N)\)-optimal |
| 6720.b4 | 6720e2 | [0, -1, 0, -1681, 27025] | [2, 2] | 4096 | |
| 6720.b3 | 6720e3 | [0, -1, 0, -2401, 2401] | [2, 2] | 8192 | |
| 6720.b1 | 6720e4 | [0, -1, 0, -26881, 1705345] | [2] | 8192 | |
| 6720.b2 | 6720e5 | [0, -1, 0, -25921, -1592255] | [2] | 16384 | |
| 6720.b6 | 6720e6 | [0, -1, 0, 9599, 9601] | [2] | 16384 |
Rank
sage: E.rank()
The elliptic curves in class 6720e have rank \(1\).
Modular form 6720.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
