| Curve name |
$X_{185f}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{185}$ |
| Curves that $X_{185f}$ minimally covers |
|
| Curves that minimally cover $X_{185f}$ |
|
| Curves that minimally cover $X_{185f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -442368t^{24} - 6856704t^{20} - 7050240t^{16} - 2681856t^{12} -
440640t^{8} - 26784t^{4} - 108\]
\[B(t) = 113246208t^{36} - 3482320896t^{32} - 10022289408t^{28} -
9314500608t^{24} - 4238770176t^{20} - 1059692544t^{16} - 145539072t^{12} -
9787392t^{8} - 212544t^{4} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 216033x - 38556063$, with conductor $4800$ |
| Generic density of odd order reductions |
$109/896$ |