| Curve name |
$X_{233}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{78}$ |
| Curves that $X_{233}$ minimally covers |
$X_{78}$, $X_{119}$, $X_{122}$ |
| Curves that minimally cover $X_{233}$ |
$X_{472}$, $X_{473}$, $X_{233a}$, $X_{233b}$, $X_{233c}$, $X_{233d}$, $X_{233e}$, $X_{233f}$, $X_{233g}$, $X_{233h}$ |
| Curves that minimally cover $X_{233}$ and have infinitely many rational
points. |
$X_{233a}$, $X_{233b}$, $X_{233c}$, $X_{233d}$, $X_{233e}$, $X_{233f}$, $X_{233g}$, $X_{233h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{233}) = \mathbb{Q}(f_{233}), f_{78} =
\frac{f_{233}}{f_{233}^{2} + \frac{1}{8}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 249696x + 48087270$, with conductor $306$ |
| Generic density of odd order reductions |
$635/5376$ |