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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 331056.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
331056.bc1 | 331056bc2 | \([0, 0, 0, -56991, 4938010]\) | \(61918288/3971\) | \(1312875342054144\) | \([2]\) | \(1520640\) | \(1.6514\) | |
331056.bc2 | 331056bc1 | \([0, 0, 0, 2904, 326095]\) | \(131072/2299\) | \(-47505357771696\) | \([2]\) | \(760320\) | \(1.3048\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 331056.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 331056.bc do not have complex multiplication.Modular form 331056.2.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.