| Curve name |
$X_{55}$ |
| Index |
$16$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 5 \\ 7 & 2 \end{matrix}\right],
\left[ \begin{matrix} 6 & 7 \\ 5 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$1$ |
$X_{1}$ |
| $4$ |
$4$ |
$X_{7}$ |
|
| Meaning/Special name |
$X_{ns}^{+}(8)$ |
| Chosen covering |
$X_{7}$ |
| Curves that $X_{55}$ minimally covers |
$X_{7}$ |
| Curves that minimally cover $X_{55}$ |
$X_{179}$, $X_{180}$, $X_{253}$, $X_{441}$ |
| Curves that minimally cover $X_{55}$ and have infinitely many rational
points. |
|
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{55}) = \mathbb{Q}(f_{55}), f_{7} =
\frac{\frac{1}{8}f_{55}^{4} - \frac{1}{8}f_{55}^{2} + \frac{1}{32}}{f_{55}^{4} -
f_{55}^{2} + f_{55} - \frac{1}{4}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 1995x + 58727$, with conductor $3388$ |
| Generic density of odd order reductions |
$2867/5376$ |