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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 361998.cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 361998.cn1 | 361998cn5 | \([1, -1, 1, -20865996716, 1160135576072295]\) | \(285531136548675601769470657/17941034271597192\) | \(63129902409069798855119112\) | \([2]\) | \(566231040\) | \(4.4109\) | |
| 361998.cn2 | 361998cn3 | \([1, -1, 1, -1306605956, 18054925839591]\) | \(70108386184777836280897/552468975892674624\) | \(1943996762068408238803020864\) | \([2, 2]\) | \(283115520\) | \(4.0643\) | |
| 361998.cn3 | 361998cn6 | \([1, -1, 1, -445050716, 41508527204967]\) | \(-2770540998624539614657/209924951154647363208\) | \(-738672112153645155443250945288\) | \([2]\) | \(566231040\) | \(4.4109\) | |
| 361998.cn4 | 361998cn2 | \([1, -1, 1, -137991236, -156765956889]\) | \(82582985847542515777/44772582831427584\) | \(157543246501941525923303424\) | \([2, 2]\) | \(141557760\) | \(3.7178\) | |
| 361998.cn5 | 361998cn1 | \([1, -1, 1, -106841156, -424507124505]\) | \(38331145780597164097/55468445663232\) | \(195179247109865107095552\) | \([2]\) | \(70778880\) | \(3.3712\) | \(\Gamma_0(N)\)-optimal |
| 361998.cn6 | 361998cn4 | \([1, -1, 1, 532222204, -1233396826905]\) | \(4738217997934888496063/2928751705237796928\) | \(-10305526790323608961784966208\) | \([2]\) | \(283115520\) | \(4.0643\) |
Rank
sage: E.rank()
The elliptic curves in class 361998.cn have rank \(1\).
Complex multiplication
The elliptic curves in class 361998.cn do not have complex multiplication.Modular form 361998.2.a.cn
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
