| Curve name |
$X_{84b}$ |
| 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 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{84}$ |
| Curves that $X_{84b}$ minimally covers |
|
| Curves that minimally cover $X_{84b}$ |
|
| Curves that minimally cover $X_{84b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{12} - 648t^{10} + 108t^{8} + 4752t^{6} + 7020t^{4} + 2808t^{2} -
108\]
\[B(t) = 432t^{18} + 3888t^{16} + 28512t^{14} + 127008t^{12} + 285120t^{10} +
300672t^{8} + 102816t^{6} - 44064t^{4} - 29808t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 8033x + 2375937$, with conductor $4800$ |
| Generic density of odd order reductions |
$691/5376$ |