| Curve name |
$X_{84}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{36}$ |
| Curves that $X_{84}$ minimally covers |
$X_{36}$ |
| Curves that minimally cover $X_{84}$ |
$X_{195}$, $X_{199}$, $X_{201}$, $X_{206}$, $X_{227}$, $X_{228}$, $X_{336}$, $X_{346}$, $X_{84a}$, $X_{84b}$, $X_{84c}$, $X_{84d}$, $X_{84e}$, $X_{84f}$, $X_{84g}$, $X_{84h}$, $X_{84i}$, $X_{84j}$, $X_{84k}$, $X_{84l}$, $X_{84m}$, $X_{84n}$, $X_{84o}$, $X_{84p}$ |
| Curves that minimally cover $X_{84}$ and have infinitely many rational
points. |
$X_{195}$, $X_{199}$, $X_{227}$, $X_{228}$, $X_{84a}$, $X_{84b}$, $X_{84c}$, $X_{84d}$, $X_{84e}$, $X_{84f}$, $X_{84g}$, $X_{84h}$, $X_{84i}$, $X_{84j}$, $X_{84k}$, $X_{84l}$, $X_{84m}$, $X_{84n}$, $X_{84o}$, $X_{84p}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{84}) = \mathbb{Q}(f_{84}), f_{36} =
\frac{-4}{f_{84}^{2} + 1}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 1989x - 458150$, with conductor $1989$ |
| Generic density of odd order reductions |
$83/672$ |