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SageMath
E = EllipticCurve("gt1")
E.isogeny_class()
Elliptic curves in class 485520.gt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485520.gt1 | 485520gt4 | \([0, 1, 0, -837040, -295006972]\) | \(2624033547076/324135\) | \(8011602870082560\) | \([2]\) | \(7864320\) | \(2.0749\) | |
485520.gt2 | 485520gt2 | \([0, 1, 0, -56740, -3799012]\) | \(3269383504/893025\) | \(5518195854393600\) | \([2, 2]\) | \(3932160\) | \(1.7283\) | |
485520.gt3 | 485520gt1 | \([0, 1, 0, -20615, 1085088]\) | \(2508888064/118125\) | \(45620005410000\) | \([2]\) | \(1966080\) | \(1.3817\) | \(\Gamma_0(N)\)-optimal* |
485520.gt4 | 485520gt3 | \([0, 1, 0, 145560, -24595452]\) | \(13799183324/18600435\) | \(-459744546040335360\) | \([2]\) | \(7864320\) | \(2.0749\) |
Rank
sage: E.rank()
The elliptic curves in class 485520.gt have rank \(1\).
Complex multiplication
The elliptic curves in class 485520.gt do not have complex multiplication.Modular form 485520.2.a.gt
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.