| Curve name |
$X_{233c}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{233}$ |
| Curves that $X_{233c}$ minimally covers |
|
| Curves that minimally cover $X_{233c}$ |
|
| Curves that minimally cover $X_{233c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{16} - 56170119168t^{14} - 102261325824t^{12} -
56877907968t^{10} - 11355586560t^{8} - 888717312t^{6} - 24966144t^{4} -
214272t^{2} - 108\]
\[B(t) = 29686813949952t^{24} - 1825739057922048t^{22} - 10033215402147840t^{20}
- 15574212840259584t^{18} - 10655833994035200t^{16} - 3557047085826048t^{14} -
611852501385216t^{12} - 55578860716032t^{10} - 2601521971200t^{8} -
59410907136t^{6} - 598026240t^{4} - 1700352t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 1775617x - 910101503$, with conductor $3264$ |
| Generic density of odd order reductions |
$109/896$ |