| Curve name |
$X_{234}$ |
| 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} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{78}$ |
| Curves that $X_{234}$ minimally covers |
$X_{78}$, $X_{120}$, $X_{121}$ |
| Curves that minimally cover $X_{234}$ |
$X_{471}$, $X_{474}$, $X_{234a}$, $X_{234b}$, $X_{234c}$, $X_{234d}$, $X_{234e}$, $X_{234f}$, $X_{234g}$, $X_{234h}$ |
| Curves that minimally cover $X_{234}$ and have infinitely many rational
points. |
$X_{234a}$, $X_{234b}$, $X_{234c}$, $X_{234d}$, $X_{234e}$, $X_{234f}$, $X_{234g}$, $X_{234h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{234}) = \mathbb{Q}(f_{234}), f_{78} =
\frac{2f_{234}^{2} + 8}{f_{234}^{2} + 4f_{234} - 4}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 9608x - 365712$, with conductor $600$ |
| Generic density of odd order reductions |
$635/5376$ |