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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 159936s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 159936.gv5 | 159936s1 | \([0, 1, 0, -220285249, 1256774022335]\) | \(38331145780597164097/55468445663232\) | \(1710701193155990406561792\) | \([2]\) | \(35389440\) | \(3.5521\) | \(\Gamma_0(N)\)-optimal |
| 159936.gv4 | 159936s2 | \([0, 1, 0, -284510529, 464118461631]\) | \(82582985847542515777/44772582831427584\) | \(1380830307296116429295714304\) | \([2, 2]\) | \(70778880\) | \(3.8987\) | |
| 159936.gv6 | 159936s3 | \([0, 1, 0, 1097336511, 3651486844095]\) | \(4738217997934888496063/2928751705237796928\) | \(-90325571172963862651147911168\) | \([2]\) | \(141557760\) | \(4.2452\) | |
| 159936.gv2 | 159936s4 | \([0, 1, 0, -2693962049, -53452178201409]\) | \(70108386184777836280897/552468975892674624\) | \(17038684335583337339676524544\) | \([2, 2]\) | \(141557760\) | \(4.2452\) | |
| 159936.gv3 | 159936s5 | \([0, 1, 0, -917606209, -122887441088833]\) | \(-2770540998624539614657/209924951154647363208\) | \(-6474291105862282807622498254848\) | \([2]\) | \(283115520\) | \(4.5918\) | |
| 159936.gv1 | 159936s6 | \([0, 1, 0, -43021542209, -3434621350524225]\) | \(285531136548675601769470657/17941034271597192\) | \(553319069389720922779287552\) | \([2]\) | \(283115520\) | \(4.5918\) |
Rank
sage: E.rank()
The elliptic curves in class 159936s have rank \(0\).
Complex multiplication
The elliptic curves in class 159936s do not have complex multiplication.Modular form 159936.2.a.s
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.
