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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 400710.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
400710.bf1 | 400710bf3 | \([1, 1, 1, -115685526, 98549325099]\) | \(3639478711331685826729/2016912141902025000\) | \(94887408615377781809025000\) | \([2]\) | \(165888000\) | \(3.6750\) | \(\Gamma_0(N)\)-optimal* |
400710.bf2 | 400710bf2 | \([1, 1, 1, -70560526, -226819974901]\) | \(825824067562227826729/5613755625000000\) | \(264104079096830625000000\) | \([2, 2]\) | \(82944000\) | \(3.3284\) | \(\Gamma_0(N)\)-optimal* |
400710.bf3 | 400710bf1 | \([1, 1, 1, -70445006, -227603754997]\) | \(821774646379511057449/38361600000\) | \(1804755268569600000\) | \([2]\) | \(41472000\) | \(2.9819\) | \(\Gamma_0(N)\)-optimal* |
400710.bf4 | 400710bf4 | \([1, 1, 1, -27283846, -502025038357]\) | \(-47744008200656797609/2286529541015625000\) | \(-107571796689605712890625000\) | \([2]\) | \(165888000\) | \(3.6750\) |
Rank
sage: E.rank()
The elliptic curves in class 400710.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 400710.bf do not have complex multiplication.Modular form 400710.2.a.bf
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.