| Curve name |
$X_{43}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 5 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{11}$ |
| Curves that $X_{43}$ minimally covers |
$X_{11}$ |
| Curves that minimally cover $X_{43}$ |
$X_{64}$, $X_{67}$, $X_{71}$, $X_{74}$, $X_{77}$, $X_{82}$, $X_{91}$, $X_{92}$, $X_{127}$, $X_{137}$, $X_{139}$, $X_{144}$ |
| Curves that minimally cover $X_{43}$ and have infinitely many rational
points. |
$X_{64}$, $X_{67}$, $X_{71}$, $X_{74}$, $X_{77}$, $X_{82}$, $X_{91}$, $X_{92}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{43}) = \mathbb{Q}(f_{43}), f_{11} =
-2f_{43}^{2} + 8\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 12x + 36$, with conductor $1176$ |
| Generic density of odd order reductions |
$25/112$ |