| Curve name |
$X_{117b}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{117}$ |
| Curves that $X_{117b}$ minimally covers |
|
| Curves that minimally cover $X_{117b}$ |
|
| Curves that minimally cover $X_{117b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -432t^{16} + 1944t^{12} - 1323t^{8} + 324t^{4} - 27\]
\[B(t) = -3456t^{24} - 23328t^{20} + 39528t^{16} - 23814t^{12} + 6885t^{8} -
972t^{4} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 12328367x + 16665315108$, with conductor $142296$ |
| Generic density of odd order reductions |
$307/2688$ |