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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 180999e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 180999.f4 | 180999e1 | \([1, -1, 1, -2500556, -1521333498]\) | \(491411892194497/78897\) | \(277618326511617\) | \([2]\) | \(2064384\) | \(2.1739\) | \(\Gamma_0(N)\)-optimal |
| 180999.f3 | 180999e2 | \([1, -1, 1, -2508161, -1511608224]\) | \(495909170514577/6224736609\) | \(21903253106787046449\) | \([2, 2]\) | \(4128768\) | \(2.5205\) | |
| 180999.f2 | 180999e3 | \([1, -1, 1, -4706006, 1539000636]\) | \(3275619238041697/1605271262049\) | \(5648538238047514826289\) | \([2, 2]\) | \(8257536\) | \(2.8671\) | |
| 180999.f5 | 180999e4 | \([1, -1, 1, -431996, -3939890808]\) | \(-2533811507137/1904381781393\) | \(-6701031511838684639073\) | \([2]\) | \(8257536\) | \(2.8671\) | |
| 180999.f1 | 180999e5 | \([1, -1, 1, -61720691, 186517444650]\) | \(7389727131216686257/6115533215337\) | \(21518994346655338262457\) | \([2]\) | \(16515072\) | \(3.2137\) | |
| 180999.f6 | 180999e6 | \([1, -1, 1, 17143159, 11773149522]\) | \(158346567380527343/108665074944153\) | \(-382364554498335792179433\) | \([2]\) | \(16515072\) | \(3.2137\) |
Rank
sage: E.rank()
The elliptic curves in class 180999e have rank \(1\).
Complex multiplication
The elliptic curves in class 180999e do not have complex multiplication.Modular form 180999.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
