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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 12342bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12342.ba2 | 12342bf1 | \([1, 0, 0, -66129, -6410007]\) | \(18052771191337/444958272\) | \(788270721302592\) | \([2]\) | \(80640\) | \(1.6421\) | \(\Gamma_0(N)\)-optimal |
12342.ba1 | 12342bf2 | \([1, 0, 0, -148409, 12695409]\) | \(204055591784617/78708537864\) | \(139436976046885704\) | \([2]\) | \(161280\) | \(1.9887\) |
Rank
sage: E.rank()
The elliptic curves in class 12342bf have rank \(1\).
Complex multiplication
The elliptic curves in class 12342bf do not have complex multiplication.Modular form 12342.2.a.bf
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 Cremona 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 Cremona labels.