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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 493680.ex
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
493680.ex1 | 493680ex3 | \([0, 1, 0, -23171056, 42922866644]\) | \(189602977175292169/1402500\) | \(10176980183040000\) | \([2]\) | \(23592960\) | \(2.6666\) | \(\Gamma_0(N)\)-optimal* |
493680.ex2 | 493680ex4 | \([0, 1, 0, -2029936, 81850580]\) | \(127483771761289/73369857660\) | \(532394714751005736960\) | \([2]\) | \(23592960\) | \(2.6666\) | |
493680.ex3 | 493680ex2 | \([0, 1, 0, -1449136, 669387860]\) | \(46380496070089/125888400\) | \(913485741229670400\) | \([2, 2]\) | \(11796480\) | \(2.3200\) | \(\Gamma_0(N)\)-optimal* |
493680.ex4 | 493680ex1 | \([0, 1, 0, -55216, 18706004]\) | \(-2565726409/19388160\) | \(-140686574050344960\) | \([2]\) | \(5898240\) | \(1.9734\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 493680.ex have rank \(1\).
Complex multiplication
The elliptic curves in class 493680.ex do not have complex multiplication.Modular form 493680.2.a.ex
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.