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SageMath
E = EllipticCurve("gr1")
E.isogeny_class()
Elliptic curves in class 136710.gr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
136710.gr1 | 136710w2 | \([1, -1, 1, -887522, -112540431]\) | \(901456690969801/457629750000\) | \(39249128511699750000\) | \([2]\) | \(5529600\) | \(2.4522\) | |
136710.gr2 | 136710w1 | \([1, -1, 1, 206158, -13671759]\) | \(11298232190519/7472736000\) | \(-640907579977056000\) | \([2]\) | \(2764800\) | \(2.1056\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 136710.gr have rank \(0\).
Complex multiplication
The elliptic curves in class 136710.gr do not have complex multiplication.Modular form 136710.2.a.gr
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.