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SageMath
sage: E = EllipticCurve("c1")
sage: E.isogeny_class()
Elliptic curves in class 5775c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 5775.q6 | 5775c1 | [1, 1, 0, 875, -2891000] | [2] | 23040 | \(\Gamma_0(N)\)-optimal |
| 5775.q5 | 5775c2 | [1, 1, 0, -299250, -62015625] | [2, 2] | 46080 | |
| 5775.q2 | 5775c3 | [1, 1, 0, -4764375, -4004721000] | [2, 2] | 92160 | |
| 5775.q4 | 5775c4 | [1, 1, 0, -636125, 102716250] | [2] | 92160 | |
| 5775.q1 | 5775c5 | [1, 1, 0, -76230000, -256206911625] | [2] | 184320 | |
| 5775.q3 | 5775c6 | [1, 1, 0, -4740750, -4046371875] | [2] | 184320 |
Rank
sage: E.rank()
The elliptic curves in class 5775c have rank \(1\).
Complex multiplication
The elliptic curves in class 5775c do not have complex multiplication.Modular form 5775.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
