| Curve name |
$X_{117p}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{117}$ |
| Curves that $X_{117p}$ minimally covers |
|
| Curves that minimally cover $X_{117p}$ |
|
| Curves that minimally cover $X_{117p}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{12} + 108t^{10} + 405t^{8} - 432t^{6} + 108t^{2} - 27\]
\[B(t) = 432t^{18} - 648t^{16} + 3564t^{14} - 4914t^{12} - 162t^{10} + 3483t^{8}
- 1512t^{6} - 324t^{4} + 324t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 101887x - 12483848$, with conductor $12936$ |
| Generic density of odd order reductions |
$691/5376$ |