| Curve name |
$X_{205}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 1 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{205}$ minimally covers |
$X_{85}$, $X_{86}$, $X_{92}$ |
| Curves that minimally cover $X_{205}$ |
$X_{444}$, $X_{453}$, $X_{476}$, $X_{478}$, $X_{205a}$, $X_{205b}$, $X_{205c}$, $X_{205d}$, $X_{205e}$, $X_{205f}$, $X_{205g}$, $X_{205h}$, $X_{205i}$, $X_{205j}$, $X_{205k}$, $X_{205l}$ |
| Curves that minimally cover $X_{205}$ and have infinitely many rational
points. |
$X_{205a}$, $X_{205b}$, $X_{205c}$, $X_{205d}$, $X_{205e}$, $X_{205f}$, $X_{205g}$, $X_{205h}$, $X_{205i}$, $X_{205j}$, $X_{205k}$, $X_{205l}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{205}) = \mathbb{Q}(f_{205}), f_{85} =
\frac{f_{205}^{2} + 1}{f_{205}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 27540x - 2745500$, with conductor $1530$ |
| Generic density of odd order reductions |
$25/224$ |