| Curve name |
$X_{192b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 14 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192b}$ minimally covers |
|
| Curves that minimally cover $X_{192b}$ |
|
| Curves that minimally cover $X_{192b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{26} - 216t^{25} - 25488t^{24} - 201312t^{23} - 719712t^{22} -
3328128t^{21} - 5412096t^{20} + 9580032t^{19} - 8729856t^{18} + 79958016t^{17} +
64806912t^{16} + 4866048t^{15} + 421576704t^{14} - 19464192t^{13} +
1036910592t^{12} - 5117313024t^{11} - 2234843136t^{10} - 9809952768t^{9} -
22167945216t^{8} + 54528049152t^{7} - 47167045632t^{6} + 52772732928t^{5} -
26726105088t^{4} + 905969664t^{3} - 452984832t^{2}\]
\[B(t) = 54t^{39} + 648t^{38} - 108864t^{37} - 1331424t^{36} - 15310944t^{35} -
132212736t^{34} - 525256704t^{33} - 1266057216t^{32} - 3700131840t^{31} -
4078854144t^{30} + 5130584064t^{29} + 35186835456t^{28} + 239444066304t^{27} +
63616057344t^{26} + 284488630272t^{25} - 747594842112t^{24} -
5970290540544t^{23} + 3552222117888t^{22} + 14208888471552t^{20} +
95524648648704t^{19} - 47846069895168t^{18} - 72829089349632t^{17} +
65142842720256t^{16} - 980762895581184t^{15} + 576501112111104t^{14} -
336237957218304t^{13} - 1069247140724736t^{12} + 3879869444259840t^{11} -
5310228845297664t^{10} + 8812345178456064t^{9} - 8872646519291904t^{8} +
4110000234430464t^{7} - 1429605634277376t^{6} + 467567319711744t^{5} +
11132555231232t^{4} - 3710851743744t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 3001250x - 2002500000$, with conductor $1050$ |
| Generic density of odd order reductions |
$1091/10752$ |