| Curve name |
$X_{252}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 5 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
$X_{s}^{+}(8)$ |
| Chosen covering |
$X_{69}$ |
| Curves that $X_{252}$ minimally covers |
$X_{69}$ |
| Curves that minimally cover $X_{252}$ |
$X_{536}$, $X_{539}$, $X_{542}$, $X_{543}$, $X_{568}$, $X_{569}$, $X_{576}$, $X_{581}$ |
| Curves that minimally cover $X_{252}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 - x\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(0 : 1 : 0)$ |
\[ \infty \]
|
| $(-1 : 0 : 1)$ |
\[-3375 \,\,(\text{CM by }-7)\]
|
| $(0 : 0 : 1)$ |
\[-3375 \,\,(\text{CM by }-7)\]
|
| $(1 : 0 : 1)$ |
\[ \infty \]
|
|
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |