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SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 436800.ga
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
436800.ga1 | 436800ga4 | \([0, -1, 0, -2497633, -1518456863]\) | \(841356017734178/1404585\) | \(2876590080000000\) | \([2]\) | \(7864320\) | \(2.2285\) | |
436800.ga2 | 436800ga3 | \([0, -1, 0, -409633, 69791137]\) | \(3711757787138/1124589375\) | \(2303159040000000000\) | \([2]\) | \(7864320\) | \(2.2285\) | \(\Gamma_0(N)\)-optimal* |
436800.ga3 | 436800ga2 | \([0, -1, 0, -157633, -23196863]\) | \(423026849956/16769025\) | \(17171481600000000\) | \([2, 2]\) | \(3932160\) | \(1.8819\) | \(\Gamma_0(N)\)-optimal* |
436800.ga4 | 436800ga1 | \([0, -1, 0, 4367, -1326863]\) | \(35969456/2985255\) | \(-764225280000000\) | \([2]\) | \(1966080\) | \(1.5353\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 436800.ga have rank \(0\).
Complex multiplication
The elliptic curves in class 436800.ga do not have complex multiplication.Modular form 436800.2.a.ga
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.