| Curve name |
$X_{112}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 10 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 13 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 12 & 15 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{45}$ |
| Curves that $X_{112}$ minimally covers |
$X_{45}$ |
| Curves that minimally cover $X_{112}$ |
$X_{210}$, $X_{216}$, $X_{220}$, $X_{231}$, $X_{310}$, $X_{323}$, $X_{326}$, $X_{351}$, $X_{361}$, $X_{385}$, $X_{396}$, $X_{403}$ |
| Curves that minimally cover $X_{112}$ and have infinitely many rational
points. |
$X_{210}$, $X_{216}$, $X_{220}$, $X_{231}$, $X_{323}$, $X_{326}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{112}) = \mathbb{Q}(f_{112}), f_{45} =
-f_{112}^{2} + 4\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 275x + 10933$, with conductor $4606$ |
| Generic density of odd order reductions |
$85091/344064$ |