| Curve name |
$X_{204}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{62}$ |
| Curves that $X_{204}$ minimally covers |
$X_{62}$, $X_{87}$, $X_{98}$ |
| Curves that minimally cover $X_{204}$ |
$X_{448}$, $X_{455}$, $X_{456}$, $X_{464}$, $X_{204a}$, $X_{204b}$, $X_{204c}$, $X_{204d}$, $X_{204e}$, $X_{204f}$, $X_{204g}$, $X_{204h}$ |
| Curves that minimally cover $X_{204}$ and have infinitely many rational
points. |
$X_{204a}$, $X_{204b}$, $X_{204c}$, $X_{204d}$, $X_{204e}$, $X_{204f}$, $X_{204g}$, $X_{204h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{204}) = \mathbb{Q}(f_{204}), f_{62} =
\frac{2f_{204}^{2} + 8}{f_{204}^{2} + 4f_{204} - 4}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 608x - 5712$, with conductor $600$ |
| Generic density of odd order reductions |
$635/5376$ |