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SageMath
E = EllipticCurve("fo1")
E.isogeny_class()
Elliptic curves in class 499800.fo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
499800.fo1 | 499800fo3 | \([0, 1, 0, -666808, 190253888]\) | \(17418812548/1753941\) | \(3301590475344000000\) | \([2]\) | \(7864320\) | \(2.2895\) | \(\Gamma_0(N)\)-optimal* |
499800.fo2 | 499800fo2 | \([0, 1, 0, -152308, -19662112]\) | \(830321872/127449\) | \(59976989604000000\) | \([2, 2]\) | \(3932160\) | \(1.9430\) | \(\Gamma_0(N)\)-optimal* |
499800.fo3 | 499800fo1 | \([0, 1, 0, -146183, -21560862]\) | \(11745974272/357\) | \(10500173250000\) | \([2]\) | \(1966080\) | \(1.5964\) | \(\Gamma_0(N)\)-optimal* |
499800.fo4 | 499800fo4 | \([0, 1, 0, 264192, -107960112]\) | \(1083360092/3306177\) | \(-6223494685968000000\) | \([2]\) | \(7864320\) | \(2.2895\) |
Rank
sage: E.rank()
The elliptic curves in class 499800.fo have rank \(1\).
Complex multiplication
The elliptic curves in class 499800.fo do not have complex multiplication.Modular form 499800.2.a.fo
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.