| Curve name |
$X_{189g}$ |
| 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} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 12 & 15 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 6 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{189}$ |
| Curves that $X_{189g}$ minimally covers |
|
| Curves that minimally cover $X_{189g}$ |
|
| Curves that minimally cover $X_{189g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{26} - 1188t^{25} - 23868t^{24} - 287712t^{23} - 2382912t^{22} -
15697152t^{21} - 97804800t^{20} - 602228736t^{19} - 3165585408t^{18} -
11728060416t^{17} - 20960722944t^{16} + 50033590272t^{15} + 492550225920t^{14} +
1704978284544t^{13} + 3005441114112t^{12} + 244158824448t^{11} -
14363696037888t^{10} - 45311166775296t^{9} - 85311539380224t^{8} -
122414620999680t^{7} - 158725885132800t^{6} - 202125455917056t^{5} -
230768592814080t^{4} - 203169132969984t^{3} - 122458107543552t^{2} -
44530220924928t - 7421703487488\]
\[B(t) = -54t^{39} - 3564t^{38} - 110808t^{37} - 2150928t^{36} - 28631232t^{35}
- 262414080t^{34} - 1433465856t^{33} + 203461632t^{32} + 95798329344t^{31} +
1119878479872t^{30} + 7908547166208t^{29} + 38634407460864t^{28} +
126787892281344t^{27} + 201115922595840t^{26} - 518215553777664t^{25} -
5016170797203456t^{24} - 19872910248247296t^{23} - 51908887229497344t^{22} -
91108353660420096t^{21} - 51825871417245696t^{20} + 457619337935585280t^{19} +
2827144407067656192t^{18} + 10302681684658618368t^{17} +
25528863944714747904t^{16} + 36529876903334510592t^{15} -
11637127164532359168t^{14} - 230689789166449852416t^{13} -
723700103713558364160t^{12} - 1420493385482694033408t^{11} -
1956539635174559711232t^{10} - 1833396830954066018304t^{9} -
859300741913637814272t^{8} + 551975244079363522560t^{7} +
1637458159016104427520t^{6} + 1902570432380616572928t^{5} +
1459531070837857714176t^{4} + 792327289642546102272t^{3} +
297669920970680303616t^{2} + 70039981404865953792t + 7782220156096217088\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 188750x - 30937500$, with conductor $1050$ |
| Generic density of odd order reductions |
$1091/10752$ |