| Curve name |
$X_{86}$ |
| 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} 7 & 0 \\ 0 & 3 \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 $X_{86}$ minimally covers |
$X_{36}$ |
| Curves that minimally cover $X_{86}$ |
$X_{184}$, $X_{197}$, $X_{201}$, $X_{205}$, $X_{212}$, $X_{229}$, $X_{337}$, $X_{345}$, $X_{86a}$, $X_{86b}$, $X_{86c}$, $X_{86d}$, $X_{86e}$, $X_{86f}$, $X_{86g}$, $X_{86h}$, $X_{86i}$, $X_{86j}$, $X_{86k}$, $X_{86l}$, $X_{86m}$, $X_{86n}$, $X_{86o}$, $X_{86p}$ |
| Curves that minimally cover $X_{86}$ and have infinitely many rational
points. |
$X_{197}$, $X_{205}$, $X_{212}$, $X_{229}$, $X_{86a}$, $X_{86b}$, $X_{86c}$, $X_{86d}$, $X_{86e}$, $X_{86f}$, $X_{86g}$, $X_{86h}$, $X_{86i}$, $X_{86j}$, $X_{86k}$, $X_{86l}$, $X_{86m}$, $X_{86n}$, $X_{86o}$, $X_{86p}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{86}) = \mathbb{Q}(f_{86}), f_{36} =
\frac{-8}{f_{86}^{2} + 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 34884x + 2462449$, with conductor $1989$ |
| Generic density of odd order reductions |
$83/672$ |