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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 58800gk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
58800.f1 | 58800gk1 | \([0, -1, 0, -427408, 61477312]\) | \(393349474783/153600000\) | \(3371827200000000000\) | \([2]\) | \(1290240\) | \(2.2539\) | \(\Gamma_0(N)\)-optimal |
58800.f2 | 58800gk2 | \([0, -1, 0, 1364592, 441381312]\) | \(12801408679457/11250000000\) | \(-246960000000000000000\) | \([2]\) | \(2580480\) | \(2.6005\) |
Rank
sage: E.rank()
The elliptic curves in class 58800gk have rank \(1\).
Complex multiplication
The elliptic curves in class 58800gk do not have complex multiplication.Modular form 58800.2.a.gk
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.