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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 198e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 198.c3 | 198e1 | \([1, -1, 0, -405, -2187]\) | \(10091699281/2737152\) | \(1995383808\) | \([2]\) | \(160\) | \(0.49244\) | \(\Gamma_0(N)\)-optimal |
| 198.c4 | 198e2 | \([1, -1, 0, 1035, -15147]\) | \(168105213359/228637728\) | \(-166676903712\) | \([2]\) | \(320\) | \(0.83901\) | |
| 198.c1 | 198e3 | \([1, -1, 0, -90585, 10516473]\) | \(112763292123580561/1932612\) | \(1408874148\) | \([2]\) | \(800\) | \(1.2972\) | |
| 198.c2 | 198e4 | \([1, -1, 0, -90495, 10538343]\) | \(-112427521449300721/466873642818\) | \(-340350885614322\) | \([2]\) | \(1600\) | \(1.6437\) |
Rank
sage: E.rank()
The elliptic curves in class 198e have rank \(0\).
Complex multiplication
The elliptic curves in class 198e do not have complex multiplication.Modular form 198.2.a.e
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.
