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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 8112.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8112.s1 | 8112bh4 | \([0, 1, 0, -187984, 31307732]\) | \(37159393753/1053\) | \(20818451976192\) | \([4]\) | \(43008\) | \(1.6573\) | |
| 8112.s2 | 8112bh3 | \([0, 1, 0, -52784, -4244460]\) | \(822656953/85683\) | \(1694005147840512\) | \([2]\) | \(43008\) | \(1.6573\) | |
| 8112.s3 | 8112bh2 | \([0, 1, 0, -12224, 444276]\) | \(10218313/1521\) | \(30071097298944\) | \([2, 2]\) | \(21504\) | \(1.3107\) | |
| 8112.s4 | 8112bh1 | \([0, 1, 0, 1296, 38676]\) | \(12167/39\) | \(-771053776896\) | \([2]\) | \(10752\) | \(0.96411\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8112.s have rank \(1\).
Complex multiplication
The elliptic curves in class 8112.s do not have complex multiplication.Modular form 8112.2.a.s
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
