| Curve name |
$X_{200b}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{200}$ |
| Curves that $X_{200b}$ minimally covers |
|
| Curves that minimally cover $X_{200b}$ |
|
| Curves that minimally cover $X_{200b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -6912t^{16} + 27648t^{14} - 435456t^{12} + 822528t^{10} - 721440t^{8} +
205632t^{6} - 27216t^{4} + 432t^{2} - 27\]
\[B(t) = 221184t^{24} - 1327104t^{22} - 25546752t^{20} + 110702592t^{18} -
314316288t^{16} + 432470016t^{14} - 285562368t^{12} + 108117504t^{10} -
19644768t^{8} + 1729728t^{6} - 99792t^{4} - 1296t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 104x + 101$, with conductor $42$ |
| Generic density of odd order reductions |
$53/896$ |