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SageMath
E = EllipticCurve("gr1")
E.isogeny_class()
Elliptic curves in class 288990gr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
288990.gr4 | 288990gr1 | \([1, -1, 1, 14198503, 43783417721]\) | \(89962967236397039/287450726400000\) | \(-1011465450114917990400000\) | \([2]\) | \(41472000\) | \(3.2893\) | \(\Gamma_0(N)\)-optimal |
288990.gr3 | 288990gr2 | \([1, -1, 1, -133764377, 513062487929]\) | \(75224183150104868881/11219310000000000\) | \(39477877065224910000000000\) | \([2]\) | \(82944000\) | \(3.6359\) | |
288990.gr2 | 288990gr3 | \([1, -1, 1, -5021528297, 136963897027481]\) | \(-3979640234041473454886161/1471455901872240\) | \(-5177676274299572719094640\) | \([2]\) | \(207360000\) | \(4.0940\) | |
288990.gr1 | 288990gr4 | \([1, -1, 1, -80344459877, 8765627517933329]\) | \(16300610738133468173382620881/2228489100\) | \(7841482117081505100\) | \([2]\) | \(414720000\) | \(4.4406\) |
Rank
sage: E.rank()
The elliptic curves in class 288990gr have rank \(1\).
Complex multiplication
The elliptic curves in class 288990gr do not have complex multiplication.Modular form 288990.2.a.gr
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.