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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1110.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1110.g1 | 1110f3 | \([1, 0, 1, -320459, -14401618]\) | \(3639478711331685826729/2016912141902025000\) | \(2016912141902025000\) | \([2]\) | \(23040\) | \(2.2028\) | |
1110.g2 | 1110f2 | \([1, 0, 1, -195459, 33048382]\) | \(825824067562227826729/5613755625000000\) | \(5613755625000000\) | \([2, 2]\) | \(11520\) | \(1.8562\) | |
1110.g3 | 1110f1 | \([1, 0, 1, -195139, 33162686]\) | \(821774646379511057449/38361600000\) | \(38361600000\) | \([2]\) | \(5760\) | \(1.5096\) | \(\Gamma_0(N)\)-optimal |
1110.g4 | 1110f4 | \([1, 0, 1, -75579, 73184206]\) | \(-47744008200656797609/2286529541015625000\) | \(-2286529541015625000\) | \([2]\) | \(23040\) | \(2.2028\) |
Rank
sage: E.rank()
The elliptic curves in class 1110.g have rank \(0\).
Complex multiplication
The elliptic curves in class 1110.g do not have complex multiplication.Modular form 1110.2.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.