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SageMath
E = EllipticCurve("cq1")
E.isogeny_class()
Elliptic curves in class 25410cq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25410.cq1 | 25410cq1 | \([1, 0, 0, -20391, 942921]\) | \(529278808969/88704000\) | \(157144546944000\) | \([2]\) | \(115200\) | \(1.4451\) | \(\Gamma_0(N)\)-optimal |
25410.cq2 | 25410cq2 | \([1, 0, 0, 37689, 5345385]\) | \(3342032927351/8893500000\) | \(-15755377753500000\) | \([2]\) | \(230400\) | \(1.7917\) |
Rank
sage: E.rank()
The elliptic curves in class 25410cq have rank \(1\).
Complex multiplication
The elliptic curves in class 25410cq do not have complex multiplication.Modular form 25410.2.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.