Show commands:
SageMath
E = EllipticCurve("fe1")
E.isogeny_class()
Elliptic curves in class 69696.fe
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69696.fe1 | 69696gk4 | \([0, 0, 0, -24534444, 46774853840]\) | \(4824238966273/66\) | \(22344338231525376\) | \([2]\) | \(2949120\) | \(2.6930\) | |
| 69696.fe2 | 69696gk2 | \([0, 0, 0, -1534764, 729494480]\) | \(1180932193/4356\) | \(1474726323280674816\) | \([2, 2]\) | \(1474560\) | \(2.3464\) | |
| 69696.fe3 | 69696gk3 | \([0, 0, 0, -837804, 1394951888]\) | \(-192100033/2371842\) | \(-802988483026327437312\) | \([2]\) | \(2949120\) | \(2.6930\) | |
| 69696.fe4 | 69696gk1 | \([0, 0, 0, -140844, -362032]\) | \(912673/528\) | \(178754705852203008\) | \([2]\) | \(737280\) | \(1.9999\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 69696.fe have rank \(2\).
Complex multiplication
The elliptic curves in class 69696.fe do not have complex multiplication.Modular form 69696.2.a.fe
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
