Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 265650.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 265650.r1 | 265650r6 | \([1, 1, 0, -1721927350, -27503101337750]\) | \(36136672427711016379227705697/1011258101510224722\) | \(15800907836097261281250\) | \([2]\) | \(104857600\) | \(3.7703\) | |
| 265650.r2 | 265650r3 | \([1, 1, 0, -123228100, 525435818500]\) | \(13244420128496241770842177/29965867631164664892\) | \(468216681736947888937500\) | \([2]\) | \(52428800\) | \(3.4237\) | |
| 265650.r3 | 265650r4 | \([1, 1, 0, -107757100, -428623734500]\) | \(8856076866003496152467137/46664863048067576004\) | \(729138485126055875062500\) | \([2, 2]\) | \(52428800\) | \(3.4237\) | |
| 265650.r4 | 265650r5 | \([1, 1, 0, -49434850, -890710921250]\) | \(-855073332201294509246497/21439133060285771735058\) | \(-334986454066965183360281250\) | \([2]\) | \(104857600\) | \(3.7703\) | |
| 265650.r5 | 265650r2 | \([1, 1, 0, -10516600, 1665478000]\) | \(8232463578739844255617/4687062591766850064\) | \(73235352996357032250000\) | \([2, 2]\) | \(26214400\) | \(3.0772\) | |
| 265650.r6 | 265650r1 | \([1, 1, 0, 2605400, 208936000]\) | \(125177609053596564863/73635189229502208\) | \(-1150549831710972000000\) | \([2]\) | \(13107200\) | \(2.7306\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 265650.r have rank \(0\).
Complex multiplication
The elliptic curves in class 265650.r do not have complex multiplication.Modular form 265650.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
