| Curve name |
$X_{75}$ |
| Index |
$24$ |
| 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],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{34}$ |
| Curves that $X_{75}$ minimally covers |
$X_{34}$, $X_{36}$, $X_{48}$ |
| Curves that minimally cover $X_{75}$ |
$X_{197}$, $X_{199}$, $X_{217}$, $X_{329}$, $X_{330}$, $X_{75a}$, $X_{75b}$, $X_{75c}$, $X_{75d}$, $X_{75e}$, $X_{75f}$, $X_{75g}$, $X_{75h}$, $X_{75i}$, $X_{75j}$ |
| Curves that minimally cover $X_{75}$ and have infinitely many rational
points. |
$X_{197}$, $X_{199}$, $X_{217}$, $X_{75a}$, $X_{75b}$, $X_{75c}$, $X_{75d}$, $X_{75e}$, $X_{75f}$, $X_{75g}$, $X_{75h}$, $X_{75i}$, $X_{75j}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{75}) = \mathbb{Q}(f_{75}), f_{34} =
\frac{f_{75}}{f_{75}^{2} + \frac{1}{8}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 125421x + 5866702$, with conductor $5544$ |
| Generic density of odd order reductions |
$643/5376$ |