Show commands:
SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 3648.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3648.bf1 | 3648bf3 | \([0, 1, 0, -6497, -203745]\) | \(115714886617/1539\) | \(403439616\) | \([2]\) | \(3072\) | \(0.79471\) | |
| 3648.bf2 | 3648bf2 | \([0, 1, 0, -417, -3105]\) | \(30664297/3249\) | \(851705856\) | \([2, 2]\) | \(1536\) | \(0.44814\) | |
| 3648.bf3 | 3648bf1 | \([0, 1, 0, -97, 287]\) | \(389017/57\) | \(14942208\) | \([2]\) | \(768\) | \(0.10156\) | \(\Gamma_0(N)\)-optimal |
| 3648.bf4 | 3648bf4 | \([0, 1, 0, 543, -14433]\) | \(67419143/390963\) | \(-102488604672\) | \([2]\) | \(3072\) | \(0.79471\) |
Rank
sage: E.rank()
The elliptic curves in class 3648.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 3648.bf do not have complex multiplication.Modular form 3648.2.a.bf
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.
