| Curve name |
$X_{192k}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192k}$ minimally covers |
|
| Curves that minimally cover $X_{192k}$ |
|
| Curves that minimally cover $X_{192k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{30} - 94176t^{28} + 4238784t^{26} - 21261312t^{24} -
1116343296t^{22} + 16016596992t^{20} - 99187310592t^{18} + 426839113728t^{16} -
1586996969472t^{14} + 4100248829952t^{12} - 4572542140416t^{10} -
1393381343232t^{8} + 4444687171584t^{6} - 1580011094016t^{4} -
28991029248t^{2}\]
\[B(t) = 432t^{45} - 927936t^{43} - 3773952t^{41} + 4221130752t^{39} -
171411738624t^{37} + 2781887348736t^{35} - 22056893153280t^{33} +
151068346417152t^{31} - 1788549399576576t^{29} + 18664420779491328t^{27} -
120985049792249856t^{25} + 483940199168999424t^{23} - 1194522929887444992t^{21}
+ 1831474585166413824t^{19} - 2475103787698618368t^{17} +
5782082198773432320t^{15} - 11668081234352799744t^{13} +
11503247055321563136t^{11} - 4532404632994971648t^{9} + 64836001666695168t^{7} +
255069105457987584t^{5} - 1899956092796928t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 3388291788x + 75913508557712$, with conductor $141120$ |
| Generic density of odd order reductions |
$139/1344$ |