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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 8034.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8034.g1 | 8034k3 | \([1, 0, 0, -2229, 40293]\) | \(1224802586728657/953137692\) | \(953137692\) | \([4]\) | \(9984\) | \(0.65427\) | |
8034.g2 | 8034k2 | \([1, 0, 0, -169, 329]\) | \(534003898897/258180624\) | \(258180624\) | \([2, 2]\) | \(4992\) | \(0.30770\) | |
8034.g3 | 8034k1 | \([1, 0, 0, -89, -327]\) | \(78018694417/1028352\) | \(1028352\) | \([2]\) | \(2496\) | \(-0.038873\) | \(\Gamma_0(N)\)-optimal |
8034.g4 | 8034k4 | \([1, 0, 0, 611, 2669]\) | \(25223358788783/17557937436\) | \(-17557937436\) | \([2]\) | \(9984\) | \(0.65427\) |
Rank
sage: E.rank()
The elliptic curves in class 8034.g have rank \(0\).
Complex multiplication
The elliptic curves in class 8034.g do not have complex multiplication.Modular form 8034.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.