| Curve name |
$X_{84e}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 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 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{84}$ |
| Curves that $X_{84e}$ minimally covers |
|
| Curves that minimally cover $X_{84e}$ |
|
| Curves that minimally cover $X_{84e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 1080t^{14} - 2916t^{12} + 2592t^{10} + 26460t^{8} +
49896t^{6} + 39204t^{4} + 10800t^{2} - 432\]
\[B(t) = 432t^{24} + 6480t^{22} + 57024t^{20} + 348192t^{18} + 1420416t^{16} +
3763584t^{14} + 6344352t^{12} + 6461856t^{10} + 3344976t^{8} + 114480t^{6} -
712800t^{4} - 243648t^{2} - 3456\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 18075x + 8009750$, with conductor $1800$ |
| Generic density of odd order reductions |
$691/5376$ |