| Curve name |
$X_{217a}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{217}$ |
| Curves that $X_{217a}$ minimally covers |
|
| Curves that minimally cover $X_{217a}$ |
|
| Curves that minimally cover $X_{217a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} + 53568t^{14} - 1560384t^{12} + 13886208t^{10} -
44357760t^{8} + 55544832t^{6} - 24966144t^{4} + 3428352t^{2} - 27648\]
\[B(t) = 432t^{24} + 425088t^{22} - 37376640t^{20} + 928295424t^{18} -
10162195200t^{16} + 54276231168t^{14} - 149378052096t^{12} + 217104924672t^{10}
- 162595123200t^{8} + 59410907136t^{6} - 9568419840t^{4} + 435290112t^{2} +
1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 24703x + 579807$, with conductor $1344$ |
| Generic density of odd order reductions |
$271/2688$ |