| Curve name |
$X_{119f}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{119}$ |
| Curves that $X_{119f}$ minimally covers |
|
| Curves that minimally cover $X_{119f}$ |
|
| Curves that minimally cover $X_{119f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{12} - 864t^{10} - 10800t^{8} - 65664t^{6} - 193968t^{4} -
228096t^{2} - 27648\]
\[B(t) = -54t^{18} - 2592t^{16} - 53136t^{14} - 604800t^{12} - 4153680t^{10} -
17459712t^{8} - 43182720t^{6} - 55655424t^{4} - 25878528t^{2} + 1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 115212x - 15052016$, with conductor $2880$ |
| Generic density of odd order reductions |
$41/336$ |