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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 111573bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
111573.h2 | 111573bf1 | \([1, -1, 1, -31982, -3463540]\) | \(-42180533641/36293103\) | \(-3112718663363463\) | \([2]\) | \(835584\) | \(1.6703\) | \(\Gamma_0(N)\)-optimal |
111573.h1 | 111573bf2 | \([1, -1, 1, -589847, -174170230]\) | \(264621653112601/81336717\) | \(6975934711964757\) | \([2]\) | \(1671168\) | \(2.0169\) |
Rank
sage: E.rank()
The elliptic curves in class 111573bf have rank \(0\).
Complex multiplication
The elliptic curves in class 111573bf do not have complex multiplication.Modular form 111573.2.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.