| Curve name |
$X_{84c}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{84}$ |
| Curves that $X_{84c}$ minimally covers |
|
| Curves that minimally cover $X_{84c}$ |
|
| Curves that minimally cover $X_{84c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{12} - 162t^{10} + 27t^{8} + 1188t^{6} + 1755t^{4} + 702t^{2} -
27\]
\[B(t) = 54t^{18} + 486t^{16} + 3564t^{14} + 15876t^{12} + 35640t^{10} +
37584t^{8} + 12852t^{6} - 5508t^{4} - 3726t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 2008x + 295988$, with conductor $1200$ |
| Generic density of odd order reductions |
$25/224$ |