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SageMath
E = EllipticCurve("gv1")
E.isogeny_class()
Elliptic curves in class 286110gv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286110.gv8 | 286110gv1 | \([1, -1, 1, 112811818, 195116821989]\) | \(9023321954633914439/6156756739584000\) | \(-108336063510466473014784000\) | \([2]\) | \(127401984\) | \(3.6842\) | \(\Gamma_0(N)\)-optimal |
| 286110.gv7 | 286110gv2 | \([1, -1, 1, -493949462, 1628529669861]\) | \(757443433548897303481/373234243041000000\) | \(6567537157737817442841000000\) | \([2, 2]\) | \(254803968\) | \(4.0308\) | |
| 286110.gv6 | 286110gv3 | \([1, -1, 1, -2030867357, 36086192638629]\) | \(-52643812360427830814761/1504091705903677440\) | \(-26466430536178159018510909440\) | \([2]\) | \(382205952\) | \(4.2336\) | |
| 286110.gv4 | 286110gv4 | \([1, -1, 1, -6463244462, 199852102593861]\) | \(1696892787277117093383481/1440538624914939000\) | \(25348132232460055788359139000\) | \([2]\) | \(509607936\) | \(4.3774\) | |
| 286110.gv5 | 286110gv5 | \([1, -1, 1, -4232834942, -104863902125691]\) | \(476646772170172569823801/5862293314453125000\) | \(103154600334995380423828125000\) | \([2]\) | \(509607936\) | \(4.3774\) | |
| 286110.gv3 | 286110gv6 | \([1, -1, 1, -32713511837, 2277404279671461]\) | \(220031146443748723000125481/172266701724057600\) | \(3031254462065540415101337600\) | \([2, 2]\) | \(764411904\) | \(4.5801\) | |
| 286110.gv1 | 286110gv7 | \([1, -1, 1, -523416091037, 145753538849896101]\) | \(901247067798311192691198986281/552431869440\) | \(9720750165110696701440\) | \([2]\) | \(1528823808\) | \(4.9267\) | |
| 286110.gv2 | 286110gv8 | \([1, -1, 1, -32933244317, 2245259087814309]\) | \(224494757451893010998773801/6152490825146276160000\) | \(108260999252285843234785124160000\) | \([2]\) | \(1528823808\) | \(4.9267\) |
Rank
sage: E.rank()
The elliptic curves in class 286110gv have rank \(1\).
Complex multiplication
The elliptic curves in class 286110gv do not have complex multiplication.Modular form 286110.2.a.gv
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
