| Curve name |
$X_{214d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 9 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{214}$ |
| Curves that $X_{214d}$ minimally covers |
|
| Curves that minimally cover $X_{214d}$ |
|
| Curves that minimally cover $X_{214d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -324t^{16} - 5184t^{15} - 48384t^{14} - 314496t^{13} - 1366848t^{12} -
3822336t^{11} - 6773760t^{10} - 7976448t^{9} - 9044352t^{8} - 15952896t^{7} -
27095040t^{6} - 30578688t^{5} - 21869568t^{4} - 10063872t^{3} - 3096576t^{2} -
663552t - 82944\]
\[B(t) = 31104t^{23} + 715392t^{22} + 7679232t^{21} + 51093504t^{20} +
239500800t^{19} + 869654016t^{18} + 2645001216t^{17} + 6948716544t^{16} +
15132192768t^{15} + 24725606400t^{14} + 24631713792t^{13} - 49263427584t^{11} -
98902425600t^{10} - 121057542144t^{9} - 111179464704t^{8} - 84640038912t^{7} -
55657857024t^{6} - 30656102400t^{5} - 13079937024t^{4} - 3931766784t^{3} -
732561408t^{2} - 63700992t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 1713028x - 677521152$, with conductor $168640$ |
| Generic density of odd order reductions |
$13411/86016$ |