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SageMath
E = EllipticCurve("hv1")
E.isogeny_class()
Elliptic curves in class 221760hv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.nf6 | 221760hv1 | \([0, 0, 0, 80088, 3781384]\) | \(76102438406144/52315569075\) | \(-39053363052211200\) | \([2]\) | \(1572864\) | \(1.8722\) | \(\Gamma_0(N)\)-optimal |
| 221760.nf5 | 221760hv2 | \([0, 0, 0, -352092, 31613776]\) | \(404151985581136/197735855625\) | \(2361744404490240000\) | \([2, 2]\) | \(3145728\) | \(2.2188\) | |
| 221760.nf2 | 221760hv3 | \([0, 0, 0, -4620972, 3820671664]\) | \(228410605013945764/187597265625\) | \(8962598937600000000\) | \([2, 2]\) | \(6291456\) | \(2.5654\) | |
| 221760.nf4 | 221760hv4 | \([0, 0, 0, -2998092, -1976171024]\) | \(62380825826921284/787768887675\) | \(37636244708725555200\) | \([2]\) | \(6291456\) | \(2.5654\) | |
| 221760.nf1 | 221760hv5 | \([0, 0, 0, -73920972, 244624311664]\) | \(467508233804095622882/315748125\) | \(30170203176960000\) | \([2]\) | \(12582912\) | \(2.9120\) | |
| 221760.nf3 | 221760hv6 | \([0, 0, 0, -3623052, 5516736496]\) | \(-55043996611705922/105743408203125\) | \(-10103940000000000000000\) | \([2]\) | \(12582912\) | \(2.9120\) |
Rank
sage: E.rank()
The elliptic curves in class 221760hv have rank \(1\).
Complex multiplication
The elliptic curves in class 221760hv do not have complex multiplication.Modular form 221760.2.a.hv
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
