Show commands:
SageMath
E = EllipticCurve("bq1")
E.isogeny_class()
Elliptic curves in class 141570bq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141570.de6 | 141570bq1 | \([1, -1, 1, 16312, 317067]\) | \(371694959/249600\) | \(-322350405062400\) | \([2]\) | \(655360\) | \(1.4726\) | \(\Gamma_0(N)\)-optimal |
| 141570.de5 | 141570bq2 | \([1, -1, 1, -70808, 2686731]\) | \(30400540561/15210000\) | \(19643227808490000\) | \([2, 2]\) | \(1310720\) | \(1.8192\) | |
| 141570.de2 | 141570bq3 | \([1, -1, 1, -920228, 339736587]\) | \(66730743078481/60937500\) | \(78698829360937500\) | \([2]\) | \(2621440\) | \(2.1658\) | |
| 141570.de3 | 141570bq4 | \([1, -1, 1, -615308, -183750069]\) | \(19948814692561/231344100\) | \(298773494967132900\) | \([2, 2]\) | \(2621440\) | \(2.1658\) | |
| 141570.de4 | 141570bq5 | \([1, -1, 1, -125258, -468763149]\) | \(-168288035761/73415764890\) | \(-94814108775069808410\) | \([2]\) | \(5242880\) | \(2.5123\) | |
| 141570.de1 | 141570bq6 | \([1, -1, 1, -9817358, -11837226189]\) | \(81025909800741361/11088090\) | \(14319913072389210\) | \([2]\) | \(5242880\) | \(2.5123\) |
Rank
sage: E.rank()
The elliptic curves in class 141570bq have rank \(2\).
Complex multiplication
The elliptic curves in class 141570bq do not have complex multiplication.Modular form 141570.2.a.bq
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 Cremona 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 Cremona labels.
