| Curve name |
$X_{234b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{234}$ |
| Curves that $X_{234b}$ minimally covers |
|
| Curves that minimally cover $X_{234b}$ |
|
| Curves that minimally cover $X_{234b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -31131t^{16} - 413856t^{15} - 2554848t^{14} - 9991296t^{13} -
26526528t^{12} - 52738560t^{11} - 84865536t^{10} - 77580288t^{9} -
169468416t^{8} + 310321152t^{7} - 1357848576t^{6} + 3375267840t^{5} -
6790791168t^{4} + 10231087104t^{3} - 10464657408t^{2} + 6780616704t -
2040201216\]
\[B(t) = 2114154t^{24} + 42158880t^{23} + 400357728t^{22} + 2437513344t^{21} +
10602010944t^{20} + 35081786880t^{19} + 91919826432t^{18} + 197139142656t^{17} +
344702449152t^{16} + 507370881024t^{15} + 640018464768t^{14} +
539658878976t^{13} + 1287773061120t^{12} - 2158635515904t^{11} +
10240295436288t^{10} - 32471736385536t^{9} + 88243826982912t^{8} -
201870482079744t^{7} + 376503609065472t^{6} - 574779996241920t^{5} +
694813389225984t^{4} - 638979498049536t^{3} + 419805504995328t^{2} -
176827159019520t + 35469618315264\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 384x + 2772$, with conductor $48$ |
| Generic density of odd order reductions |
$53/896$ |