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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 13650.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13650.g1 | 13650b3 | \([1, 1, 0, -1052900, -28461750]\) | \(8261629364934163009/4759323790524030\) | \(74364434226937968750\) | \([2]\) | \(491520\) | \(2.5025\) | |
13650.g2 | 13650b2 | \([1, 1, 0, -749150, -249288000]\) | \(2975849362756797409/8263842596100\) | \(129122540564062500\) | \([2, 2]\) | \(245760\) | \(2.1560\) | |
13650.g3 | 13650b1 | \([1, 1, 0, -748650, -249637500]\) | \(2969894891179808929/22997520\) | \(359336250000\) | \([2]\) | \(122880\) | \(1.8094\) | \(\Gamma_0(N)\)-optimal |
13650.g4 | 13650b4 | \([1, 1, 0, -453400, -447736250]\) | \(-659704930833045889/5156082432978750\) | \(-80563788015292968750\) | \([2]\) | \(491520\) | \(2.5025\) |
Rank
sage: E.rank()
The elliptic curves in class 13650.g have rank \(0\).
Complex multiplication
The elliptic curves in class 13650.g do not have complex multiplication.Modular form 13650.2.a.g
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.