| Curve name |
$X_{34}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 0 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{13}$ |
| Curves that $X_{34}$ minimally covers |
$X_{13}$ |
| Curves that minimally cover $X_{34}$ |
$X_{75}$, $X_{79}$, $X_{94}$, $X_{100}$, $X_{34a}$, $X_{34b}$, $X_{34c}$, $X_{34d}$, $X_{34e}$, $X_{34f}$, $X_{34g}$, $X_{34h}$ |
| Curves that minimally cover $X_{34}$ and have infinitely many rational
points. |
$X_{75}$, $X_{79}$, $X_{94}$, $X_{100}$, $X_{34a}$, $X_{34b}$, $X_{34c}$, $X_{34d}$, $X_{34e}$, $X_{34f}$, $X_{34g}$, $X_{34h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{34}) = \mathbb{Q}(f_{34}), f_{13} =
8f_{34}^{2} - 8\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 58x - 284$, with conductor $350$ |
| Generic density of odd order reductions |
$513/3584$ |