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SageMath
E = EllipticCurve("cq1")
E.isogeny_class()
Elliptic curves in class 32448cq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32448.bu1 | 32448cq1 | \([0, -1, 0, -440301, -105501267]\) | \(1909913257984/129730653\) | \(641213525479707648\) | \([2]\) | \(645120\) | \(2.1654\) | \(\Gamma_0(N)\)-optimal |
32448.bu2 | 32448cq2 | \([0, -1, 0, 381039, -454570767]\) | \(77366117936/1172914587\) | \(-92756945875155075072\) | \([2]\) | \(1290240\) | \(2.5120\) |
Rank
sage: E.rank()
The elliptic curves in class 32448cq have rank \(0\).
Complex multiplication
The elliptic curves in class 32448cq do not have complex multiplication.Modular form 32448.2.a.cq
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.