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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 333270.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.bc1 | 333270bc2 | \([1, -1, 0, -1755, -18225]\) | \(67419143/22050\) | \(195577833150\) | \([2]\) | \(393216\) | \(0.86972\) | |
333270.bc2 | 333270bc1 | \([1, -1, 0, 315, -2079]\) | \(389017/420\) | \(-3725292060\) | \([2]\) | \(196608\) | \(0.52315\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 333270.bc have rank \(2\).
Complex multiplication
The elliptic curves in class 333270.bc do not have complex multiplication.Modular form 333270.2.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.