| Curve name |
$X_{29}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 6 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{11}$ |
| Curves that $X_{29}$ minimally covers |
$X_{11}$, $X_{16}$, $X_{19}$ |
| Curves that minimally cover $X_{29}$ |
$X_{76}$, $X_{77}$, $X_{83}$, $X_{125}$, $X_{133}$ |
| Curves that minimally cover $X_{29}$ and have infinitely many rational
points. |
$X_{76}$, $X_{77}$, $X_{83}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{29}) = \mathbb{Q}(f_{29}), f_{11} =
\frac{8f_{29}^{2} - 16}{f_{29}^{2} + 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 3699x - 118827$, with conductor $17664$ |
| Generic density of odd order reductions |
$2659/10752$ |