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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 5775.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5775.g1 | 5775o3 | \([1, 0, 0, -517563, -40647258]\) | \(981281029968144361/522287841796875\) | \(8160747528076171875\) | \([2]\) | \(110592\) | \(2.3202\) | |
5775.g2 | 5775o2 | \([1, 0, 0, -406188, -99564633]\) | \(474334834335054841/607815140625\) | \(9497111572265625\) | \([2, 2]\) | \(55296\) | \(1.9736\) | |
5775.g3 | 5775o1 | \([1, 0, 0, -406063, -99629008]\) | \(473897054735271721/779625\) | \(12181640625\) | \([2]\) | \(27648\) | \(1.6270\) | \(\Gamma_0(N)\)-optimal |
5775.g4 | 5775o4 | \([1, 0, 0, -296813, -154361508]\) | \(-185077034913624841/551466161890875\) | \(-8616658779544921875\) | \([2]\) | \(110592\) | \(2.3202\) |
Rank
sage: E.rank()
The elliptic curves in class 5775.g have rank \(0\).
Complex multiplication
The elliptic curves in class 5775.g do not have complex multiplication.Modular form 5775.2.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.