| Curve name |
$X_{193e}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 0 & 9 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{193}$ |
| Curves that $X_{193e}$ minimally covers |
|
| Curves that minimally cover $X_{193e}$ |
|
| Curves that minimally cover $X_{193e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{26} + 905969664t^{25} + 452984832t^{24} - 1585446912t^{23}
+ 396361728t^{22} + 169869312t^{21} - 84934656t^{20} + 382205952t^{19} -
536150016t^{18} + 828112896t^{17} - 449445888t^{16} + 192872448t^{15} -
162349056t^{14} - 48218112t^{13} - 28090368t^{12} - 12939264t^{11} -
2094336t^{10} - 373248t^{9} - 20736t^{8} - 10368t^{7} + 6048t^{6} + 6048t^{5} +
432t^{4} - 216t^{3} - 27t^{2}\]
\[B(t) = 3710851743744t^{39} - 11132555231232t^{38} + 26903675142144t^{36} -
18786186952704t^{35} - 11132555231232t^{34} + 12987981103104t^{33} -
8349416423424t^{32} - 521838526464t^{31} + 24526410743808t^{30} -
18090402250752t^{29} - 8479876055040t^{28} + 12295820279808t^{27} -
13089449705472t^{26} + 13241652609024t^{25} - 5120540540928t^{24} +
3302599163904t^{23} - 1566534795264t^{22} - 391633698816t^{20} -
206412447744t^{19} - 80008445952t^{18} - 51725205504t^{17} - 12782665728t^{16} -
3001909248t^{15} - 517570560t^{14} + 276037632t^{13} + 93560832t^{12} +
497664t^{11} - 1990656t^{10} - 774144t^{9} - 165888t^{8} + 69984t^{7} +
25056t^{6} - 648t^{4} - 54t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 1310776x - 338871802$, with conductor $7350$ |
| Generic density of odd order reductions |
$1091/10752$ |