| Curve name |
$X_{199e}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{199}$ |
| Curves that $X_{199e}$ minimally covers |
|
| Curves that minimally cover $X_{199e}$ |
|
| Curves that minimally cover $X_{199e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} + 12528t^{14} - 79056t^{12} - 257472t^{10} - 306720t^{8} -
1029888t^{6} - 1264896t^{4} + 801792t^{2} - 6912\]
\[B(t) = -54t^{24} - 55728t^{22} + 2494800t^{20} - 1100736t^{18} - 4740768t^{16}
+ 120310272t^{14} + 404006400t^{12} + 481241088t^{10} - 75852288t^{8} -
70447104t^{6} + 638668800t^{4} - 57065472t^{2} - 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 26304x + 1980288$, with conductor $816$ |
| Generic density of odd order reductions |
$215/2688$ |