| Curve name |
$X_{64}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{23}$ |
| Curves that $X_{64}$ minimally covers |
$X_{23}$, $X_{43}$, $X_{47}$ |
| Curves that minimally cover $X_{64}$ |
$X_{191}$, $X_{196}$, $X_{255}$, $X_{257}$, $X_{263}$, $X_{278}$ |
| Curves that minimally cover $X_{64}$ and have infinitely many rational
points. |
$X_{191}$, $X_{196}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{64}) = \mathbb{Q}(f_{64}), f_{23} =
\frac{f_{64}^{2} + \frac{1}{4}}{f_{64}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 204083x + 35462018$, with conductor $424830$ |
| Generic density of odd order reductions |
$271/1344$ |