Show commands:
SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 82800.dl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82800.dl1 | 82800du4 | \([0, 0, 0, -397440075, 3049689312250]\) | \(148809678420065817601/20700\) | \(965779200000000\) | \([2]\) | \(7077888\) | \(3.2002\) | |
| 82800.dl2 | 82800du6 | \([0, 0, 0, -92988075, -295638659750]\) | \(1905890658841300321/293666194803750\) | \(13701289984763760000000000\) | \([2]\) | \(14155776\) | \(3.5468\) | |
| 82800.dl3 | 82800du3 | \([0, 0, 0, -25488075, 45033840250]\) | \(39248884582600321/3935264062500\) | \(183603680100000000000000\) | \([2, 2]\) | \(7077888\) | \(3.2002\) | |
| 82800.dl4 | 82800du2 | \([0, 0, 0, -24840075, 47651112250]\) | \(36330796409313601/428490000\) | \(19991629440000000000\) | \([2, 2]\) | \(3538944\) | \(2.8536\) | |
| 82800.dl5 | 82800du1 | \([0, 0, 0, -1512075, 785160250]\) | \(-8194759433281/965779200\) | \(-45059394355200000000\) | \([2]\) | \(1769472\) | \(2.5070\) | \(\Gamma_0(N)\)-optimal |
| 82800.dl6 | 82800du5 | \([0, 0, 0, 31643925, 218200932250]\) | \(75108181893694559/484313964843750\) | \(-22596152343750000000000000\) | \([2]\) | \(14155776\) | \(3.5468\) |
Rank
sage: E.rank()
The elliptic curves in class 82800.dl have rank \(0\).
Complex multiplication
The elliptic curves in class 82800.dl do not have complex multiplication.Modular form 82800.2.a.dl
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
