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SageMath
E = EllipticCurve("ne1")
E.isogeny_class()
Elliptic curves in class 286650.ne
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.ne1 | 286650ne4 | \([1, -1, 1, -5542130, -3006594503]\) | \(520300455507/193072360\) | \(6985856554951876875000\) | \([2]\) | \(23887872\) | \(2.8915\) | |
| 286650.ne2 | 286650ne2 | \([1, -1, 1, -4899005, -4172365753]\) | \(261984288445803/42250\) | \(2097001511718750\) | \([2]\) | \(7962624\) | \(2.3422\) | |
| 286650.ne3 | 286650ne1 | \([1, -1, 1, -305255, -65553253]\) | \(-63378025803/812500\) | \(-40326952148437500\) | \([2]\) | \(3981312\) | \(1.9956\) | \(\Gamma_0(N)\)-optimal |
| 286650.ne4 | 286650ne3 | \([1, -1, 1, 1072870, -334134503]\) | \(3774555693/3515200\) | \(-127189013289975000000\) | \([2]\) | \(11943936\) | \(2.5449\) |
Rank
sage: E.rank()
The elliptic curves in class 286650.ne have rank \(0\).
Complex multiplication
The elliptic curves in class 286650.ne do not have complex multiplication.Modular form 286650.2.a.ne
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
