Show commands for:
SageMath
sage: E = EllipticCurve("eu1")
sage: E.isogeny_class()
Elliptic curves in class 28224.eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 28224.eu1 | 28224cb4 | [0, 0, 0, -37933644, 89925934448] | [2] | 1179648 | |
| 28224.eu2 | 28224cb6 | [0, 0, 0, -25797324, -49948258192] | [2] | 2359296 | |
| 28224.eu3 | 28224cb3 | [0, 0, 0, -2935884, 685259120] | [2, 2] | 1179648 | |
| 28224.eu4 | 28224cb2 | [0, 0, 0, -2371404, 1404406640] | [2, 2] | 589824 | |
| 28224.eu5 | 28224cb1 | [0, 0, 0, -113484, 32494448] | [2] | 294912 | \(\Gamma_0(N)\)-optimal |
| 28224.eu6 | 28224cb5 | [0, 0, 0, 10893876, 5293335152] | [2] | 2359296 |
Rank
sage: E.rank()
The elliptic curves in class 28224.eu have rank \(1\).
Complex multiplication
The elliptic curves in class 28224.eu do not have complex multiplication.Modular form 28224.2.a.eu
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.
