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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 32368.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32368.bc1 | 32368bc2 | \([0, -1, 0, -594280, 33692016]\) | \(234770924809/130960928\) | \(12947777273462915072\) | \([2]\) | \(1105920\) | \(2.3576\) | |
32368.bc2 | 32368bc1 | \([0, -1, 0, 145560, 4098416]\) | \(3449795831/2071552\) | \(-204809131364712448\) | \([2]\) | \(552960\) | \(2.0110\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32368.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 32368.bc do not have complex multiplication.Modular form 32368.2.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.