| Curve name |
$X_{84g}$ |
| 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 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{84}$ |
| Curves that $X_{84g}$ minimally covers |
|
| Curves that minimally cover $X_{84g}$ |
|
| Curves that minimally cover $X_{84g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 270t^{14} - 729t^{12} + 648t^{10} + 6615t^{8} + 12474t^{6}
+ 9801t^{4} + 2700t^{2} - 108\]
\[B(t) = -54t^{24} - 810t^{22} - 7128t^{20} - 43524t^{18} - 177552t^{16} -
470448t^{14} - 793044t^{12} - 807732t^{10} - 418122t^{8} - 14310t^{6} +
89100t^{4} + 30456t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 72300x - 64078000$, with conductor $14400$ |
| Generic density of odd order reductions |
$89/672$ |