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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 2112.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2112.bc1 | 2112o1 | \([0, 1, 0, -41, 87]\) | \(1906624/33\) | \(135168\) | \([2]\) | \(384\) | \(-0.21850\) | \(\Gamma_0(N)\)-optimal |
2112.bc2 | 2112o2 | \([0, 1, 0, -1, 287]\) | \(-8/1089\) | \(-35684352\) | \([2]\) | \(768\) | \(0.12807\) |
Rank
sage: E.rank()
The elliptic curves in class 2112.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 2112.bc do not have complex multiplication.Modular form 2112.2.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.