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SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 397488.gg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 397488.gg1 | 397488gg3 | \([0, 1, 0, -259231184, 1606082912916]\) | \(828279937799497/193444524\) | \(449950503548138592190464\) | \([4]\) | \(74317824\) | \(3.5292\) | |
| 397488.gg2 | 397488gg2 | \([0, 1, 0, -18088464, 18881529876]\) | \(281397674377/96589584\) | \(224666643746944372506624\) | \([2, 2]\) | \(37158912\) | \(3.1826\) | |
| 397488.gg3 | 397488gg1 | \([0, 1, 0, -7488784, -7672788460]\) | \(19968681097/628992\) | \(1463030647110748864512\) | \([2]\) | \(18579456\) | \(2.8361\) | \(\Gamma_0(N)\)-optimal |
| 397488.gg4 | 397488gg4 | \([0, 1, 0, 53459376, 131326115220]\) | \(7264187703863/7406095788\) | \(-17226522933967095821549568\) | \([2]\) | \(74317824\) | \(3.5292\) |
Rank
sage: E.rank()
The elliptic curves in class 397488.gg have rank \(0\).
Complex multiplication
The elliptic curves in class 397488.gg do not have complex multiplication.Modular form 397488.2.a.gg
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.
