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SageMath
E = EllipticCurve("gs1")
E.isogeny_class()
Elliptic curves in class 92400.gs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
92400.gs1 | 92400ig4 | \([0, 1, 0, -2661911208, -52860773198412]\) | \(260744057755293612689909/8504954620259328\) | \(68039636962074624000000000\) | \([2]\) | \(46080000\) | \(4.0509\) | |
92400.gs2 | 92400ig3 | \([0, 1, 0, -173591208, -750375758412]\) | \(72313087342699809269/11447096545640448\) | \(91576772365123584000000000\) | \([2]\) | \(23040000\) | \(3.7043\) | |
92400.gs3 | 92400ig2 | \([0, 1, 0, -47101208, 123265821588]\) | \(1444540994277943589/15251205665388\) | \(122009645323104000000000\) | \([2]\) | \(9216000\) | \(3.2462\) | |
92400.gs4 | 92400ig1 | \([0, 1, 0, -46981208, 123930861588]\) | \(1433528304665250149/162339408\) | \(1298715264000000000\) | \([2]\) | \(4608000\) | \(2.8996\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 92400.gs have rank \(1\).
Complex multiplication
The elliptic curves in class 92400.gs do not have complex multiplication.Modular form 92400.2.a.gs
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.