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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 17787h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17787.s6 | 17787h1 | \([1, 1, 0, 5806, -35097]\) | \(103823/63\) | \(-13130609945607\) | \([2]\) | \(30720\) | \(1.2064\) | \(\Gamma_0(N)\)-optimal |
| 17787.s5 | 17787h2 | \([1, 1, 0, -23839, -313760]\) | \(7189057/3969\) | \(827228426573241\) | \([2, 2]\) | \(61440\) | \(1.5530\) | |
| 17787.s2 | 17787h3 | \([1, 1, 0, -290644, -60344885]\) | \(13027640977/21609\) | \(4503799211343201\) | \([2, 2]\) | \(122880\) | \(1.8995\) | |
| 17787.s3 | 17787h4 | \([1, 1, 0, -231354, 42475833]\) | \(6570725617/45927\) | \(9572214650347503\) | \([2]\) | \(122880\) | \(1.8995\) | |
| 17787.s1 | 17787h5 | \([1, 1, 0, -4648459, -3859488002]\) | \(53297461115137/147\) | \(30638089873083\) | \([2]\) | \(245760\) | \(2.2461\) | |
| 17787.s4 | 17787h6 | \([1, 1, 0, -201709, -97857668]\) | \(-4354703137/17294403\) | \(-3604540635478341867\) | \([2]\) | \(245760\) | \(2.2461\) |
Rank
sage: E.rank()
The elliptic curves in class 17787h have rank \(1\).
Complex multiplication
The elliptic curves in class 17787h do not have complex multiplication.Modular form 17787.2.a.h
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.
