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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 491970.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
491970.bd1 | 491970bd2 | \([1, 0, 1, -1064624, 148057166]\) | \(901456690969801/457629750000\) | \(67745626874097750000\) | \([2]\) | \(21626880\) | \(2.4977\) | \(\Gamma_0(N)\)-optimal* |
491970.bd2 | 491970bd1 | \([1, 0, 1, 247296, 17914702]\) | \(11298232190519/7472736000\) | \(-1106233117022304000\) | \([2]\) | \(10813440\) | \(2.1511\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 491970.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 491970.bd do not have complex multiplication.Modular form 491970.2.a.bd
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.