| Curve name |
$X_{714}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$7$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 23 & 19 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 31 & 27 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 25 & 4 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 9 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{350}$ |
| Curves that $X_{714}$ minimally covers |
$X_{350}$ |
| Curves that minimally cover $X_{714}$ |
|
| Curves that minimally cover $X_{714}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + 8x\]\[w^2 = 35048x^2y^3 - 2594064x^2y^2 - 30247840x^2y -
153541248x^2 - 1014xy^4 - 473316xy^3 + 14902128xy^2 + 8972288xy - 4153344x -
268158y^4 - 841152y^3 + 21141888y^2 - 48396544y + 175475712\] |
| 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 |