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SageMath
sage: E = EllipticCurve("m1")
sage: E.isogeny_class()
Elliptic curves in class 12705m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 12705.n6 | 12705m1 | [1, 0, 1, 4232, 30787601] | [2] | 115200 | \(\Gamma_0(N)\)-optimal |
| 12705.n5 | 12705m2 | [1, 0, 1, -1448373, 658894003] | [2, 2] | 230400 | |
| 12705.n4 | 12705m3 | [1, 0, 1, -3078848, -1096801477] | [2] | 460800 | |
| 12705.n2 | 12705m4 | [1, 0, 1, -23059578, 42619209631] | [2, 2] | 460800 | |
| 12705.n1 | 12705m5 | [1, 0, 1, -368953203, 2727722241781] | [2] | 921600 | |
| 12705.n3 | 12705m6 | [1, 0, 1, -22945233, 43062822493] | [2] | 921600 |
Rank
sage: E.rank()
The elliptic curves in class 12705m have rank \(0\).
Complex multiplication
The elliptic curves in class 12705m do not have complex multiplication.Modular form 12705.2.a.m
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
