Show commands:
SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 139638.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139638.o1 | 139638bd6 | \([1, 0, 1, -37981565, -90099554866]\) | \(2361739090258884097/5202\) | \(13346908779618\) | \([2]\) | \(6635520\) | \(2.6525\) | |
| 139638.o2 | 139638bd4 | \([1, 0, 1, -2373875, -1407920614]\) | \(576615941610337/27060804\) | \(69430619471572836\) | \([2, 2]\) | \(3317760\) | \(2.3060\) | |
| 139638.o3 | 139638bd5 | \([1, 0, 1, -2250665, -1560553162]\) | \(-491411892194497/125563633938\) | \(-322161931604735268642\) | \([2]\) | \(6635520\) | \(2.6525\) | |
| 139638.o4 | 139638bd2 | \([1, 0, 1, -156095, -19590334]\) | \(163936758817/30338064\) | \(77839172002732176\) | \([2, 2]\) | \(1658880\) | \(1.9594\) | |
| 139638.o5 | 139638bd1 | \([1, 0, 1, -46575, 3584098]\) | \(4354703137/352512\) | \(904449347889408\) | \([2]\) | \(829440\) | \(1.6128\) | \(\Gamma_0(N)\)-optimal |
| 139638.o6 | 139638bd3 | \([1, 0, 1, 309365, -113985622]\) | \(1276229915423/2927177028\) | \(-7510335404557732452\) | \([2]\) | \(3317760\) | \(2.3060\) |
Rank
sage: E.rank()
The elliptic curves in class 139638.o have rank \(0\).
Complex multiplication
The elliptic curves in class 139638.o do not have complex multiplication.Modular form 139638.2.a.o
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
