| Curve name |
$X_{202h}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{202}$ |
| Curves that $X_{202h}$ minimally covers |
|
| Curves that minimally cover $X_{202h}$ |
|
| Curves that minimally cover $X_{202h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -6912t^{16} - 801792t^{14} - 1264896t^{12} + 1029888t^{10} -
306720t^{8} + 257472t^{6} - 79056t^{4} - 12528t^{2} - 27\]
\[B(t) = -221184t^{24} + 57065472t^{22} + 638668800t^{20} + 70447104t^{18} -
75852288t^{16} - 481241088t^{14} + 404006400t^{12} - 120310272t^{10} -
4740768t^{8} + 1100736t^{6} + 2494800t^{4} + 55728t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 14624x + 669300$, with conductor $336$ |
| Generic density of odd order reductions |
$53/896$ |