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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 8550.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8550.bc1 | 8550bk2 | \([1, -1, 1, -7070, -116593]\) | \(428831641421/181752822\) | \(16562225904750\) | \([2]\) | \(21504\) | \(1.2332\) | |
8550.bc2 | 8550bk1 | \([1, -1, 1, 1480, -13993]\) | \(3936827539/3158028\) | \(-287775301500\) | \([2]\) | \(10752\) | \(0.88667\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8550.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 8550.bc do not have complex multiplication.Modular form 8550.2.a.bc
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.