| Curve name |
$X_{37}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
| $4$ |
$6$ |
$X_{9}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{9}$ |
| Curves that $X_{37}$ minimally covers |
$X_{9}$, $X_{17}$, $X_{18}$ |
| Curves that minimally cover $X_{37}$ |
$X_{123}$, $X_{37a}$, $X_{37b}$, $X_{37c}$, $X_{37d}$ |
| Curves that minimally cover $X_{37}$ and have infinitely many rational
points. |
$X_{123}$, $X_{37a}$, $X_{37b}$, $X_{37c}$, $X_{37d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{37}) = \mathbb{Q}(f_{37}), f_{9} =
\frac{8f_{37}^{2} - 1}{f_{37}^{2} + f_{37} + \frac{1}{8}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 201x + 199$, with conductor $4480$ |
| Generic density of odd order reductions |
$2659/10752$ |