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SageMath
sage: E = EllipticCurve("ep1")
sage: E.isogeny_class()
Elliptic curves in class 471510.ep
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
471510.ep1 | 471510ep5 | [1, -1, 1, -467737952, 3893723161319] | [2] | 62914560 | \(\Gamma_0(N)\)-optimal* |
471510.ep2 | 471510ep3 | [1, -1, 1, -29233652, 60844775879] | [2, 2] | 31457280 | \(\Gamma_0(N)\)-optimal* |
471510.ep3 | 471510ep6 | [1, -1, 1, -28777352, 62835704039] | [2] | 62914560 | |
471510.ep4 | 471510ep4 | [1, -1, 1, -5627732, -4005018169] | [2] | 31457280 | |
471510.ep5 | 471510ep2 | [1, -1, 1, -1855652, 919809479] | [2, 2] | 15728640 | \(\Gamma_0(N)\)-optimal* |
471510.ep6 | 471510ep1 | [1, -1, 1, 91228, 60067271] | [2] | 7864320 | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 471510.ep have rank \(1\).
Complex multiplication
The elliptic curves in class 471510.ep do not have complex multiplication.Modular form 471510.2.a.ep
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.