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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 397488gk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
397488.gk3 | 397488gk1 | \([0, 1, 0, -354384, -81216108]\) | \(4649101309/6804\) | \(7203483237629952\) | \([2]\) | \(2764800\) | \(1.9444\) | \(\Gamma_0(N)\)-optimal |
397488.gk4 | 397488gk2 | \([0, 1, 0, -252464, -128792364]\) | \(-1680914269/5786802\) | \(-6126562493604274176\) | \([2]\) | \(5529600\) | \(2.2910\) | |
397488.gk1 | 397488gk3 | \([0, 1, 0, -10597344, 13270038900]\) | \(124318741396429/51631104\) | \(54662520900107108352\) | \([2]\) | \(13824000\) | \(2.7491\) | |
397488.gk2 | 397488gk4 | \([0, 1, 0, -8966624, 17494908276]\) | \(-75306487574989/81352871712\) | \(-86129342697009390944256\) | \([2]\) | \(27648000\) | \(3.0957\) |
Rank
sage: E.rank()
The elliptic curves in class 397488gk have rank \(1\).
Complex multiplication
The elliptic curves in class 397488gk do not have complex multiplication.Modular form 397488.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.