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SageMath
E = EllipticCurve("ea1")
E.isogeny_class()
Elliptic curves in class 406560ea
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
406560.ea3 | 406560ea1 | \([0, 1, 0, -523486, 118462760]\) | \(139927692143296/27348890625\) | \(3100814593569000000\) | \([2, 2]\) | \(5898240\) | \(2.2649\) | \(\Gamma_0(N)\)-optimal* |
406560.ea1 | 406560ea2 | \([0, 1, 0, -7934736, 8599897260]\) | \(60910917333827912/3255076125\) | \(2952481748521536000\) | \([2]\) | \(11796480\) | \(2.6115\) | \(\Gamma_0(N)\)-optimal* |
406560.ea4 | 406560ea3 | \([0, 1, 0, 1077344, 702445544]\) | \(152461584507448/322998046875\) | \(-292971900375000000000\) | \([2]\) | \(11796480\) | \(2.6115\) | |
406560.ea2 | 406560ea4 | \([0, 1, 0, -2565361, -1475424865]\) | \(257307998572864/19456203375\) | \(141180318135166464000\) | \([2]\) | \(11796480\) | \(2.6115\) |
Rank
sage: E.rank()
The elliptic curves in class 406560ea have rank \(1\).
Complex multiplication
The elliptic curves in class 406560ea do not have complex multiplication.Modular form 406560.2.a.ea
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.