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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 9408cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9408.m6 | 9408cd1 | \([0, -1, 0, 3071, -15455]\) | \(103823/63\) | \(-1942981705728\) | \([2]\) | \(12288\) | \(1.0472\) | \(\Gamma_0(N)\)-optimal |
| 9408.m5 | 9408cd2 | \([0, -1, 0, -12609, -112671]\) | \(7189057/3969\) | \(122407847460864\) | \([2, 2]\) | \(24576\) | \(1.3937\) | |
| 9408.m2 | 9408cd3 | \([0, -1, 0, -153729, -23115231]\) | \(13027640977/21609\) | \(666442725064704\) | \([2, 2]\) | \(49152\) | \(1.7403\) | |
| 9408.m3 | 9408cd4 | \([0, -1, 0, -122369, 16417185]\) | \(6570725617/45927\) | \(1416433663475712\) | \([2]\) | \(49152\) | \(1.7403\) | |
| 9408.m1 | 9408cd5 | \([0, -1, 0, -2458689, -1483076895]\) | \(53297461115137/147\) | \(4533623980032\) | \([2]\) | \(98304\) | \(2.0869\) | |
| 9408.m4 | 9408cd6 | \([0, -1, 0, -106689, -37575327]\) | \(-4354703137/17294403\) | \(-533376327626784768\) | \([2]\) | \(98304\) | \(2.0869\) |
Rank
sage: E.rank()
The elliptic curves in class 9408cd have rank \(1\).
Complex multiplication
The elliptic curves in class 9408cd do not have complex multiplication.Modular form 9408.2.a.cd
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.
