| Curve name |
$X_{192f}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192f}$ minimally covers |
|
| Curves that minimally cover $X_{192f}$ |
|
| Curves that minimally cover $X_{192f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{30} - 23544t^{28} + 1059696t^{26} - 5315328t^{24} -
279085824t^{22} + 4004149248t^{20} - 24796827648t^{18} + 106709778432t^{16} -
396749242368t^{14} + 1025062207488t^{12} - 1143135535104t^{10} -
348345335808t^{8} + 1111171792896t^{6} - 395002773504t^{4} - 7247757312t^{2}\]
\[B(t) = 54t^{45} - 115992t^{43} - 471744t^{41} + 527641344t^{39} -
21426467328t^{37} + 347735918592t^{35} - 2757111644160t^{33} +
18883543302144t^{31} - 223568674947072t^{29} + 2333052597436416t^{27} -
15123131224031232t^{25} + 60492524896124928t^{23} - 149315366235930624t^{21} +
228934323145801728t^{19} - 309387973462327296t^{17} + 722760274846679040t^{15} -
1458510154294099968t^{13} + 1437905881915195392t^{11} - 566550579124371456t^{9}
+ 8104500208336896t^{7} + 31883638182248448t^{5} - 237494511599616t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 52942059x - 148255335887$, with conductor
$4410$ |
| Generic density of odd order reductions |
$271/2688$ |