| Curve name |
$X_{87}$ |
| 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} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{25}$ |
| Curves that $X_{87}$ minimally covers |
$X_{25}$ |
| Curves that minimally cover $X_{87}$ |
$X_{181}$, $X_{182}$, $X_{183}$, $X_{184}$, $X_{185}$, $X_{186}$, $X_{198}$, $X_{204}$, $X_{269}$, $X_{270}$, $X_{273}$, $X_{274}$, $X_{87a}$, $X_{87b}$, $X_{87c}$, $X_{87d}$, $X_{87e}$, $X_{87f}$, $X_{87g}$, $X_{87h}$, $X_{87i}$, $X_{87j}$, $X_{87k}$, $X_{87l}$, $X_{87m}$, $X_{87n}$, $X_{87o}$, $X_{87p}$ |
| Curves that minimally cover $X_{87}$ and have infinitely many rational
points. |
$X_{181}$, $X_{183}$, $X_{185}$, $X_{204}$, $X_{87a}$, $X_{87b}$, $X_{87c}$, $X_{87d}$, $X_{87e}$, $X_{87f}$, $X_{87g}$, $X_{87h}$, $X_{87i}$, $X_{87j}$, $X_{87k}$, $X_{87l}$, $X_{87m}$, $X_{87n}$, $X_{87o}$, $X_{87p}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{87}) = \mathbb{Q}(f_{87}), f_{25} =
-2f_{87}^{2} - 1\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 12501x + 368032$, with conductor $1989$ |
| Generic density of odd order reductions |
$83/672$ |