Show commands:
SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 6930.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6930.bf1 | 6930bi3 | \([1, -1, 1, -109652, -13886049]\) | \(200005594092187129/1027287538200\) | \(748892615347800\) | \([2]\) | \(49152\) | \(1.7001\) | |
| 6930.bf2 | 6930bi2 | \([1, -1, 1, -10652, 53151]\) | \(183337554283129/104587560000\) | \(76244331240000\) | \([2, 2]\) | \(24576\) | \(1.3535\) | |
| 6930.bf3 | 6930bi1 | \([1, -1, 1, -7772, 265119]\) | \(71210194441849/165580800\) | \(120708403200\) | \([4]\) | \(12288\) | \(1.0070\) | \(\Gamma_0(N)\)-optimal |
| 6930.bf4 | 6930bi4 | \([1, -1, 1, 42268, 391839]\) | \(11456208593737991/6725709375000\) | \(-4903042134375000\) | \([2]\) | \(49152\) | \(1.7001\) |
Rank
sage: E.rank()
The elliptic curves in class 6930.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 6930.bf do not have complex multiplication.Modular form 6930.2.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.
