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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 429429.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
429429.bg1 | 429429bg2 | \([1, 0, 0, -2142, 37335]\) | \(371694959/7203\) | \(21063063021\) | \([2]\) | \(387072\) | \(0.77359\) | \(\Gamma_0(N)\)-optimal* |
429429.bg2 | 429429bg1 | \([1, 0, 0, 3, 1728]\) | \(1/441\) | \(-1289575287\) | \([2]\) | \(193536\) | \(0.42702\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 429429.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 429429.bg do not have complex multiplication.Modular form 429429.2.a.bg
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.