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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 203280.de
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 203280.de1 | 203280cj7 | \([0, -1, 0, -496443680, 3994708607232]\) | \(1864737106103260904761/129177711985836360\) | \(937354021369201654816604160\) | \([2]\) | \(79626240\) | \(3.9234\) | |
| 203280.de2 | 203280cj4 | \([0, -1, 0, -487876880, 4147921207872]\) | \(1769857772964702379561/691787250\) | \(5019825407579136000\) | \([2]\) | \(26542080\) | \(3.3741\) | |
| 203280.de3 | 203280cj6 | \([0, -1, 0, -98014880, -298441398528]\) | \(14351050585434661561/3001282273281600\) | \(21778246145380452669849600\) | \([2, 2]\) | \(39813120\) | \(3.5768\) | |
| 203280.de4 | 203280cj3 | \([0, -1, 0, -92439200, -342034295040]\) | \(12038605770121350841/757333463040\) | \(5495449301469615882240\) | \([2]\) | \(19906560\) | \(3.2302\) | |
| 203280.de5 | 203280cj2 | \([0, -1, 0, -30496880, 64798471872]\) | \(432288716775559561/270140062500\) | \(1960221078579456000000\) | \([2, 2]\) | \(13271040\) | \(3.0275\) | |
| 203280.de6 | 203280cj5 | \([0, -1, 0, -24746960, 89973921600]\) | \(-230979395175477481/348191894531250\) | \(-2526589668834000000000000\) | \([4]\) | \(26542080\) | \(3.3741\) | |
| 203280.de7 | 203280cj1 | \([0, -1, 0, -2270000, 599256000]\) | \(178272935636041/81841914000\) | \(593870614559760384000\) | \([2]\) | \(6635520\) | \(2.6809\) | \(\Gamma_0(N)\)-optimal |
| 203280.de8 | 203280cj8 | \([0, -1, 0, 211203040, -1801735238400]\) | \(143584693754978072519/276341298967965000\) | \(-2005219196686282929623040000\) | \([4]\) | \(79626240\) | \(3.9234\) |
Rank
sage: E.rank()
The elliptic curves in class 203280.de have rank \(1\).
Complex multiplication
The elliptic curves in class 203280.de do not have complex multiplication.Modular form 203280.2.a.de
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
