| Curve name |
$X_{189n}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 12 \\ 12 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 12 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 10 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 9 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{189}$ |
| Curves that $X_{189n}$ minimally covers |
|
| Curves that minimally cover $X_{189n}$ |
|
| Curves that minimally cover $X_{189n}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{28} - 1296t^{27} - 28512t^{26} - 378432t^{25} - 3438720t^{24} -
24095232t^{23} - 151372800t^{22} - 933949440t^{21} - 5207003136t^{20} -
22124593152t^{19} - 56108187648t^{18} + 7629963264t^{17} + 751360278528t^{16} +
3409956569088t^{15} + 7880803614720t^{14} + 6404299554816t^{13} -
21463780294656t^{12} - 96076270927872t^{11} - 207459805298688t^{10} -
315741299539968t^{9} - 410223063859200t^{8} - 526709019377664t^{7} -
639658069327872t^{6} - 617856815333376t^{5} - 410049117683712t^{4} -
163277476724736t^{3} - 29686813949952t^{2}\]
\[B(t) = 54t^{42} + 3888t^{41} + 132192t^{40} + 2814912t^{39} + 41482368t^{38} +
432594432t^{37} + 2978519040t^{36} + 8032296960t^{35} - 100035772416t^{34} -
1707483856896t^{33} - 14577356242944t^{32} - 84668003647488t^{31} -
345087100846080t^{30} - 880011707940864t^{29} - 355136937394176t^{28} +
8918426978353152t^{27} + 49861062392020992t^{26} + 160979401322790912t^{25} +
353393050883457024t^{24} + 457619337935585280t^{23} - 414606971337965568t^{22} -
5830934634266886144t^{21} - 26577350261502640128t^{20} -
81399440376820924416t^{19} - 164369884682762846208t^{18} -
135847098129491951616t^{17} + 421770659303711047680t^{16} +
2127147854988841058304t^{15} + 5185422392023414996992t^{14} +
8491737859434209083392t^{13} + 9619682893088868532224t^{12} +
6583211064702674141184t^{11} + 111854215095140745216t^{10} -
6304449506767508865024t^{9} - 9232874632068527554560t^{8} -
8058975360397388808192t^{7} - 4843459269650383110144t^{6} -
1996139470038679683072t^{5} - 513626530302350327808t^{4} -
62257761248769736704t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 1698755x + 833613747$, with conductor
$3150$ |
| Generic density of odd order reductions |
$11/112$ |