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SageMath
E = EllipticCurve("da1")
E.isogeny_class()
Elliptic curves in class 84966da
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84966.cr5 | 84966da1 | \([1, 1, 1, -994725579, -12060749479815]\) | \(38331145780597164097/55468445663232\) | \(157517120697727379614728192\) | \([2]\) | \(53084160\) | \(3.9290\) | \(\Gamma_0(N)\)-optimal |
| 84966.cr4 | 84966da2 | \([1, 1, 1, -1284742859, -4454988308359]\) | \(82582985847542515777/44772582831427584\) | \(127143428114514212586055471104\) | \([2, 2]\) | \(106168320\) | \(4.2755\) | |
| 84966.cr6 | 84966da3 | \([1, 1, 1, 4955160181, -35033009165575]\) | \(4738217997934888496063/2928751705237796928\) | \(-8316954447371773411745474376768\) | \([2]\) | \(212336640\) | \(4.6221\) | |
| 84966.cr2 | 84966da4 | \([1, 1, 1, -12164922379, 512897547867641]\) | \(70108386184777836280897/552468975892674624\) | \(1568879771497199860712885089344\) | \([2, 2]\) | \(212336640\) | \(4.6221\) | |
| 84966.cr3 | 84966da5 | \([1, 1, 1, -4143565539, 1179186740968137]\) | \(-2770540998624539614657/209924951154647363208\) | \(-596136658835743544641501531901448\) | \([2]\) | \(424673280\) | \(4.9687\) | |
| 84966.cr1 | 84966da6 | \([1, 1, 1, -194269151539, 32957388273621545]\) | \(285531136548675601769470657/17941034271597192\) | \(50948246827736574799837970952\) | \([2]\) | \(424673280\) | \(4.9687\) |
Rank
sage: E.rank()
The elliptic curves in class 84966da have rank \(1\).
Complex multiplication
The elliptic curves in class 84966da do not have complex multiplication.Modular form 84966.2.a.da
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
