| Curve name |
$X_{187h}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 4 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 6 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{187}$ |
| Curves that $X_{187h}$ minimally covers |
|
| Curves that minimally cover $X_{187h}$ |
|
| Curves that minimally cover $X_{187h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} - 216t^{20} - 6480t^{16} - 48384t^{12} - 103680t^{8} -
55296t^{4} - 110592\]
\[B(t) = 54t^{36} + 648t^{32} - 25920t^{28} - 338688t^{24} - 1824768t^{20} -
7299072t^{16} - 21676032t^{12} - 26542080t^{8} + 10616832t^{4} + 14155776\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 251x - 727$, with conductor $75$ |
| Generic density of odd order reductions |
$11/112$ |