| Curve name |
$X_{66f}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{66}$ |
| Curves that $X_{66f}$ minimally covers |
|
| Curves that minimally cover $X_{66f}$ |
|
| Curves that minimally cover $X_{66f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -324t^{16} - 10368t^{15} - 152064t^{14} - 1354752t^{13} - 7769088t^{12}
- 25436160t^{11} - 11501568t^{10} + 295501824t^{9} + 1336836096t^{8} +
2364014592t^{7} - 736100352t^{6} - 13023313920t^{5} - 31822184448t^{4} -
44392513536t^{3} - 39862665216t^{2} - 21743271936t - 5435817984\]
\[B(t) = 31104t^{23} + 1430784t^{22} + 29389824t^{21} + 353009664t^{20} +
2701320192t^{19} + 13103382528t^{18} + 33342160896t^{17} - 28877783040t^{16} -
642983657472t^{15} - 2806071164928t^{14} - 6014053122048t^{13} +
48112424976384t^{11} + 179588554555392t^{10} + 329207632625664t^{9} +
118283399331840t^{8} - 1092555928240128t^{7} - 3434973109420032t^{6} -
5665079043293184t^{5} - 5922519383015424t^{4} - 3944635403599872t^{3} -
1536292621910016t^{2} - 267181325549568t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 160132x + 24630144$, with conductor $15680$ |
| Generic density of odd order reductions |
$419/2688$ |