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SageMath
E = EllipticCurve("eb1")
E.isogeny_class()
Elliptic curves in class 390390.eb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 390390.eb1 | 390390eb7 | \([1, 0, 0, -43336251, 103020562761]\) | \(1864737106103260904761/129177711985836360\) | \(623516142812642814975240\) | \([2]\) | \(63700992\) | \(3.3138\) | |
| 390390.eb2 | 390390eb4 | \([1, 0, 0, -42588426, 106972239906]\) | \(1769857772964702379561/691787250\) | \(3339124924385250\) | \([2]\) | \(21233664\) | \(2.7645\) | |
| 390390.eb3 | 390390eb6 | \([1, 0, 0, -8556051, -7698725919]\) | \(14351050585434661561/3001282273281600\) | \(14486616288216086414400\) | \([2, 2]\) | \(31850496\) | \(2.9672\) | |
| 390390.eb4 | 390390eb3 | \([1, 0, 0, -8069331, -8822951775]\) | \(12038605770121350841/757333463040\) | \(3655503975402639360\) | \([2]\) | \(15925248\) | \(2.6206\) | |
| 390390.eb5 | 390390eb2 | \([1, 0, 0, -2662176, 1670748156]\) | \(432288716775559561/270140062500\) | \(1303914484935562500\) | \([2, 2]\) | \(10616832\) | \(2.4179\) | |
| 390390.eb6 | 390390eb5 | \([1, 0, 0, -2160246, 2320145190]\) | \(-230979395175477481/348191894531250\) | \(-1680655770250488281250\) | \([2]\) | \(21233664\) | \(2.7645\) | |
| 390390.eb7 | 390390eb1 | \([1, 0, 0, -198156, 15419520]\) | \(178272935636041/81841914000\) | \(395035287072426000\) | \([2]\) | \(5308416\) | \(2.0713\) | \(\Gamma_0(N)\)-optimal |
| 390390.eb8 | 390390eb8 | \([1, 0, 0, 18436629, -46465612935]\) | \(143584693754978072519/276341298967965000\) | \(-1333846668930264173685000\) | \([2]\) | \(63700992\) | \(3.3138\) |
Rank
sage: E.rank()
The elliptic curves in class 390390.eb have rank \(0\).
Complex multiplication
The elliptic curves in class 390390.eb do not have complex multiplication.Modular form 390390.2.a.eb
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.
