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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 89232.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
89232.e1 | 89232bn2 | \([0, -1, 0, -2084, 18108]\) | \(810448/363\) | \(448545706752\) | \([2]\) | \(110592\) | \(0.93118\) | |
89232.e2 | 89232bn1 | \([0, -1, 0, 451, 1884]\) | \(131072/99\) | \(-7645665456\) | \([2]\) | \(55296\) | \(0.58461\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 89232.e have rank \(1\).
Complex multiplication
The elliptic curves in class 89232.e do not have complex multiplication.Modular form 89232.2.a.e
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.