Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 55440.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.e1 | 55440ct6 | \([0, 0, 0, -439084803, -3541365261502]\) | \(3135316978843283198764801/571725\) | \(1707161702400\) | \([2]\) | \(3932160\) | \(3.1416\) | |
| 55440.e2 | 55440ct4 | \([0, 0, 0, -27442803, -55333820302]\) | \(765458482133960722801/326869475625\) | \(976027024304640000\) | \([2, 2]\) | \(1966080\) | \(2.7950\) | |
| 55440.e3 | 55440ct5 | \([0, 0, 0, -27306723, -55909738078]\) | \(-754127868744065783521/15825714261328125\) | \(-47255329572897600000000\) | \([2]\) | \(3932160\) | \(3.1416\) | |
| 55440.e4 | 55440ct3 | \([0, 0, 0, -3664083, 1423613522]\) | \(1821931919215868881/761147600816295\) | \(2272774557675843809280\) | \([2]\) | \(1966080\) | \(2.7950\) | |
| 55440.e5 | 55440ct2 | \([0, 0, 0, -1723683, -855580318]\) | \(189674274234120481/3859869269025\) | \(11525507879400345600\) | \([2, 2]\) | \(983040\) | \(2.4484\) | |
| 55440.e6 | 55440ct1 | \([0, 0, 0, 5037, -39970222]\) | \(4733169839/231139696095\) | \(-690179434304532480\) | \([2]\) | \(491520\) | \(2.1019\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55440.e have rank \(1\).
Complex multiplication
The elliptic curves in class 55440.e do not have complex multiplication.Modular form 55440.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
