| Curve name |
$X_{197}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{75}$ |
| Curves that $X_{197}$ minimally covers |
$X_{75}$, $X_{86}$, $X_{102}$ |
| Curves that minimally cover $X_{197}$ |
$X_{467}$, $X_{481}$, $X_{197a}$, $X_{197b}$, $X_{197c}$, $X_{197d}$, $X_{197e}$, $X_{197f}$, $X_{197g}$, $X_{197h}$ |
| Curves that minimally cover $X_{197}$ and have infinitely many rational
points. |
$X_{197a}$, $X_{197b}$, $X_{197c}$, $X_{197d}$, $X_{197e}$, $X_{197f}$, $X_{197g}$, $X_{197h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{197}) = \mathbb{Q}(f_{197}), f_{75} =
\frac{\frac{1}{4}f_{197}^{2} + f_{197} - 1}{f_{197}^{2} + 4}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 392x - 21712$, with conductor $600$ |
| Generic density of odd order reductions |
$635/5376$ |