| Curve name |
$X_{230h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{230}$ |
| Curves that $X_{230h}$ minimally covers |
|
| Curves that minimally cover $X_{230h}$ |
|
| Curves that minimally cover $X_{230h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -110592t^{24} + 1105920t^{22} - 4036608t^{20} + 6248448t^{18} -
3089664t^{16} - 967680t^{14} + 946944t^{12} - 241920t^{10} - 193104t^{8} +
97632t^{6} - 15768t^{4} + 1080t^{2} - 27\]
\[B(t) = 14155776t^{36} - 212336640t^{34} + 1305870336t^{32} - 4190109696t^{30}
+ 7293763584t^{28} - 6168379392t^{26} + 959938560t^{24} + 1995964416t^{22} -
1225912320t^{20} + 302026752t^{18} - 306478080t^{16} + 124747776t^{14} +
14999040t^{12} - 24095232t^{10} + 7122816t^{8} - 1022976t^{6} + 79704t^{4} -
3240t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 197x - 2367$, with conductor $294$ |
| Generic density of odd order reductions |
$81/896$ |