Show commands for:
SageMath
sage: E = EllipticCurve("10580.c1")
sage: E.isogeny_class()
Elliptic curves in class 10580m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 10580.c3 | 10580m1 | [0, 1, 0, -705, -5192] | [2] | 5940 | \(\Gamma_0(N)\)-optimal |
| 10580.c4 | 10580m2 | [0, 1, 0, 1940, -32700] | [2] | 11880 | |
| 10580.c1 | 10580m3 | [0, 1, 0, -21865, 1236900] | [2] | 17820 | |
| 10580.c2 | 10580m4 | [0, 1, 0, -19220, 1550068] | [2] | 35640 |
Rank
sage: E.rank()
The elliptic curves in class 10580m have rank \(0\).
Modular form 10580.2.a.c
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.
