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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 58989.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58989.n1 | 58989a6 | \([1, 1, 0, -2202314, -1258878483]\) | \(53297461115137/147\) | \(3258161085963\) | \([2]\) | \(599040\) | \(2.0594\) | |
| 58989.n2 | 58989a4 | \([1, 1, 0, -137699, -19696560]\) | \(13027640977/21609\) | \(478949679636561\) | \([2, 2]\) | \(299520\) | \(1.7128\) | |
| 58989.n3 | 58989a3 | \([1, 1, 0, -109609, 13837282]\) | \(6570725617/45927\) | \(1017942613571583\) | \([2]\) | \(299520\) | \(1.7128\) | |
| 58989.n4 | 58989a5 | \([1, 1, 0, -95564, -31924137]\) | \(-4354703137/17294403\) | \(-383319393602460987\) | \([2]\) | \(599040\) | \(2.0594\) | |
| 58989.n5 | 58989a2 | \([1, 1, 0, -11294, -103785]\) | \(7189057/3969\) | \(87970349321001\) | \([2, 2]\) | \(149760\) | \(1.3662\) | |
| 58989.n6 | 58989a1 | \([1, 1, 0, 2751, -11088]\) | \(103823/63\) | \(-1396354751127\) | \([2]\) | \(74880\) | \(1.0196\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 58989.n have rank \(1\).
Complex multiplication
The elliptic curves in class 58989.n do not have complex multiplication.Modular form 58989.2.a.n
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
