| Curve name |
$X_{99}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{25}$ |
| Curves that $X_{99}$ minimally covers |
$X_{25}$ |
| Curves that minimally cover $X_{99}$ |
$X_{182}$, $X_{186}$, $X_{187}$, $X_{190}$, $X_{194}$, $X_{198}$, $X_{203}$, $X_{206}$, $X_{268}$, $X_{270}$, $X_{271}$, $X_{272}$, $X_{99a}$, $X_{99b}$, $X_{99c}$, $X_{99d}$, $X_{99e}$, $X_{99f}$, $X_{99g}$, $X_{99h}$, $X_{99i}$, $X_{99j}$, $X_{99k}$, $X_{99l}$, $X_{99m}$, $X_{99n}$, $X_{99o}$, $X_{99p}$ |
| Curves that minimally cover $X_{99}$ and have infinitely many rational
points. |
$X_{187}$, $X_{190}$, $X_{194}$, $X_{203}$, $X_{99a}$, $X_{99b}$, $X_{99c}$, $X_{99d}$, $X_{99e}$, $X_{99f}$, $X_{99g}$, $X_{99h}$, $X_{99i}$, $X_{99j}$, $X_{99k}$, $X_{99l}$, $X_{99m}$, $X_{99n}$, $X_{99o}$, $X_{99p}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{99}) = \mathbb{Q}(f_{99}), f_{25} =
-f_{99}^{2} - 1\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 4896x - 126293$, with conductor $1989$ |
| Generic density of odd order reductions |
$83/672$ |