| Curve name |
$X_{706}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$7$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 25 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 9 & 9 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 15 & 13 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 21 & 10 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{284}$ |
| Curves that $X_{706}$ minimally covers |
$X_{284}$ |
| Curves that minimally cover $X_{706}$ |
|
| Curves that minimally cover $X_{706}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + 8x\]\[w^2 = 17524x^2y^3 - 1297032x^2y^2 - 15123920x^2y -
76770624x^2 - 507xy^4 - 236658xy^3 + 7451064xy^2 + 4486144xy - 2076672x -
134079y^4 - 420576y^3 + 10570944y^2 - 24198272y + 87737856\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(57121/169 : 13652397/2197 : 1 : 0)$ |
Singular
|
| $(0 : 0 : 0 : 1)$ |
Singular
|
|
| Comments on finding rational points |
This curve admits a family of twists of etale double covers that are also
modular curves. Each of these modular curves maps to one we have already
computed that has finitely many rational points. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |