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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 85680.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
85680.g1 | 85680ds6 | \([0, 0, 0, -10330563, -12755312638]\) | \(40832710302042509761/91556816413125\) | \(273387188900528640000\) | \([2]\) | \(4194304\) | \(2.8037\) | |
85680.g2 | 85680ds4 | \([0, 0, 0, -880563, -41282638]\) | \(25288177725059761/14387797265625\) | \(42961732430400000000\) | \([2, 2]\) | \(2097152\) | \(2.4571\) | |
85680.g3 | 85680ds2 | \([0, 0, 0, -563043, 161866658]\) | \(6610905152742241/35128130625\) | \(104892035996160000\) | \([2, 2]\) | \(1048576\) | \(2.1105\) | |
85680.g4 | 85680ds1 | \([0, 0, 0, -562323, 162303122]\) | \(6585576176607121/187425\) | \(559648051200\) | \([2]\) | \(524288\) | \(1.7639\) | \(\Gamma_0(N)\)-optimal |
85680.g5 | 85680ds3 | \([0, 0, 0, -257043, 337082258]\) | \(-629004249876241/16074715228425\) | \(-47998842476633395200\) | \([2]\) | \(2097152\) | \(2.4571\) | |
85680.g6 | 85680ds5 | \([0, 0, 0, 3489117, -328807582]\) | \(1573196002879828319/926055908203125\) | \(-2765188125000000000000\) | \([2]\) | \(4194304\) | \(2.8037\) |
Rank
sage: E.rank()
The elliptic curves in class 85680.g have rank \(1\).
Complex multiplication
The elliptic curves in class 85680.g do not have complex multiplication.Modular form 85680.2.a.g
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.