| Curve name |
$X_{223}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{102}$ |
| Curves that $X_{223}$ minimally covers |
$X_{102}$, $X_{117}$, $X_{121}$ |
| Curves that minimally cover $X_{223}$ |
$X_{474}$, $X_{475}$, $X_{223a}$, $X_{223b}$, $X_{223c}$, $X_{223d}$, $X_{223e}$, $X_{223f}$, $X_{223g}$, $X_{223h}$ |
| Curves that minimally cover $X_{223}$ and have infinitely many rational
points. |
$X_{223a}$, $X_{223b}$, $X_{223c}$, $X_{223d}$, $X_{223e}$, $X_{223f}$, $X_{223g}$, $X_{223h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{223}) = \mathbb{Q}(f_{223}), f_{102} =
8f_{223}^{2} + 1\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 306x - 1836$, with conductor $306$ |
| Generic density of odd order reductions |
$635/5376$ |