Show commands:
SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 22848.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22848.bc1 | 22848bx6 | \([0, -1, 0, -877990657, 10013724179617]\) | \(285531136548675601769470657/17941034271597192\) | \(4703134488093574299648\) | \([4]\) | \(5898240\) | \(3.6189\) | |
| 22848.bc2 | 22848bx4 | \([0, -1, 0, -54978817, 155852962465]\) | \(70108386184777836280897/552468975892674624\) | \(144826427216409296633856\) | \([2, 2]\) | \(2949120\) | \(3.2723\) | |
| 22848.bc3 | 22848bx5 | \([0, -1, 0, -18726657, 358277773473]\) | \(-2770540998624539614657/209924951154647363208\) | \(-55030566395483878380797952\) | \([2]\) | \(5898240\) | \(3.6189\) | |
| 22848.bc4 | 22848bx2 | \([0, -1, 0, -5806337, -1351456095]\) | \(82582985847542515777/44772582831427584\) | \(11736863953761752580096\) | \([2, 2]\) | \(1474560\) | \(2.9257\) | |
| 22848.bc5 | 22848bx1 | \([0, -1, 0, -4495617, -3662779743]\) | \(38331145780597164097/55468445663232\) | \(14540720219942289408\) | \([2]\) | \(737280\) | \(2.5791\) | \(\Gamma_0(N)\)-optimal |
| 22848.bc6 | 22848bx3 | \([0, -1, 0, 22394623, -10652132703]\) | \(4738217997934888496063/2928751705237796928\) | \(-767754687017857037893632\) | \([2]\) | \(2949120\) | \(3.2723\) |
Rank
sage: E.rank()
The elliptic curves in class 22848.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 22848.bc do not have complex multiplication.Modular form 22848.2.a.bc
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
