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SageMath
E = EllipticCurve("gm1")
E.isogeny_class()
Elliptic curves in class 212160.gm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.gm1 | 212160l3 | \([0, 1, 0, -10632545, 13340946975]\) | \(507102228823216499929/2648775168000\) | \(694360517640192000\) | \([2]\) | \(6635520\) | \(2.6197\) | |
| 212160.gm2 | 212160l4 | \([0, 1, 0, -10448225, 13825966623]\) | \(-481184224995688814809/36713242449000000\) | \(-9624156228550656000000\) | \([2]\) | \(13271040\) | \(2.9662\) | |
| 212160.gm3 | 212160l1 | \([0, 1, 0, -186785, 1307103]\) | \(2749236527524969/1587903192720\) | \(416259294552391680\) | \([2]\) | \(2211840\) | \(2.0704\) | \(\Gamma_0(N)\)-optimal |
| 212160.gm4 | 212160l2 | \([0, 1, 0, 746335, 11198175]\) | \(175381844946241751/101691694692900\) | \(-26657867613575577600\) | \([2]\) | \(4423680\) | \(2.4169\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.gm have rank \(0\).
Complex multiplication
The elliptic curves in class 212160.gm do not have complex multiplication.Modular form 212160.2.a.gm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
