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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 106722.eg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106722.eg1 | 106722hn3 | \([1, -1, 1, -537079577, 4790912463285]\) | \(112763292123580561/1932612\) | \(293640903172022039172\) | \([2]\) | \(34560000\) | \(3.4691\) | |
| 106722.eg2 | 106722hn4 | \([1, -1, 1, -536545967, 4800906978585]\) | \(-112427521449300721/466873642818\) | \(-70936741645135982146034658\) | \([2]\) | \(69120000\) | \(3.8156\) | |
| 106722.eg3 | 106722hn1 | \([1, -1, 1, -2402357, -1044075315]\) | \(10091699281/2737152\) | \(415882642454412198912\) | \([2]\) | \(6912000\) | \(2.6643\) | \(\Gamma_0(N)\)-optimal |
| 106722.eg4 | 106722hn2 | \([1, -1, 1, 6135403, -6798525555]\) | \(168105213359/228637728\) | \(-34739196977520118990368\) | \([2]\) | \(13824000\) | \(3.0109\) |
Rank
sage: E.rank()
The elliptic curves in class 106722.eg have rank \(1\).
Complex multiplication
The elliptic curves in class 106722.eg do not have complex multiplication.Modular form 106722.2.a.eg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
