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SageMath
sage: E = EllipticCurve("f1")
sage: E.isogeny_class()
Elliptic curves in class 8085.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 8085.f1 | 8085g5 | [1, 1, 1, -149410801, 702882354698] | [2] | 368640 | |
| 8085.f2 | 8085g4 | [1, 1, 1, -9338176, 10979616248] | [2, 2] | 184320 | |
| 8085.f3 | 8085g6 | [1, 1, 1, -9291871, 11093952554] | [2] | 368640 | |
| 8085.f4 | 8085g3 | [1, 1, 1, -1246806, -283100196] | [2] | 184320 | |
| 8085.f5 | 8085g2 | [1, 1, 1, -586531, 169584344] | [2, 2] | 92160 | |
| 8085.f6 | 8085g1 | [1, 1, 1, 1714, 7934618] | [2] | 46080 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8085.f have rank \(0\).
Complex multiplication
The elliptic curves in class 8085.f do not have complex multiplication.Modular form 8085.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
