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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 166635.bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
166635.bi1 | 166635bw3 | \([1, -1, 0, -219010860, 1247572088091]\) | \(10765299591712341649/20708625\) | \(2234836769933273625\) | \([2]\) | \(17842176\) | \(3.2025\) | |
166635.bi2 | 166635bw2 | \([1, -1, 0, -13692735, 19482255216]\) | \(2630872462131649/3645140625\) | \(393376880421928265625\) | \([2, 2]\) | \(8921088\) | \(2.8560\) | |
166635.bi3 | 166635bw4 | \([1, -1, 0, -9860130, 30619038825]\) | \(-982374577874929/3183837890625\) | \(-343593936703935791015625\) | \([2]\) | \(17842176\) | \(3.2025\) | |
166635.bi4 | 166635bw1 | \([1, -1, 0, -1099890, 116978175]\) | \(1363569097969/734582625\) | \(79274807521219067625\) | \([2]\) | \(4460544\) | \(2.5094\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 166635.bi have rank \(1\).
Complex multiplication
The elliptic curves in class 166635.bi do not have complex multiplication.Modular form 166635.2.a.bi
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.