| Curve name |
$X_{188b}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{188}$ |
| Curves that $X_{188b}$ minimally covers |
|
| Curves that minimally cover $X_{188b}$ |
|
| Curves that minimally cover $X_{188b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} + 1080t^{22} - 15768t^{20} + 97632t^{18} - 296784t^{16} +
2246400t^{14} - 14812416t^{12} + 8985600t^{10} - 4748544t^{8} + 6248448t^{6} -
4036608t^{4} + 1105920t^{2} - 110592\]
\[B(t) = 54t^{36} - 3240t^{34} + 79704t^{32} - 1022976t^{30} + 6687360t^{28} -
4935168t^{26} - 300271104t^{24} + 2437890048t^{22} - 7064755200t^{20} +
6154555392t^{18} - 28259020800t^{16} + 39006240768t^{14} - 19217350656t^{12} -
1263403008t^{10} + 6847856640t^{8} - 4190109696t^{6} + 1305870336t^{4} -
212336640t^{2} + 14155776\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 4117x - 101935$, with conductor $294$ |
| Generic density of odd order reductions |
$81/896$ |