| Curve name |
$X_{105}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 13 & 10 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 15 & 13 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{39}$ |
| Curves that $X_{105}$ minimally covers |
$X_{39}$ |
| Curves that minimally cover $X_{105}$ |
$X_{214}$, $X_{221}$, $X_{224}$, $X_{232}$, $X_{282}$, $X_{288}$, $X_{301}$, $X_{302}$, $X_{388}$, $X_{389}$, $X_{390}$, $X_{394}$ |
| Curves that minimally cover $X_{105}$ and have infinitely many rational
points. |
$X_{214}$, $X_{221}$, $X_{224}$, $X_{232}$, $X_{288}$, $X_{302}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{105}) = \mathbb{Q}(f_{105}), f_{39} =
-f_{105}^{2} + 4\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 9235x + 342453$, with conductor $4606$ |
| Generic density of odd order reductions |
$85091/344064$ |