| Curve name |
$X_{189c}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 4 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 6 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{189}$ |
| Curves that $X_{189c}$ minimally covers |
|
| Curves that minimally cover $X_{189c}$ |
|
| Curves that minimally cover $X_{189c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} - 1296t^{23} - 27648t^{22} - 342144t^{21} - 2761344t^{20} -
16782336t^{19} - 99090432t^{18} - 629710848t^{17} - 3314110464t^{16} -
10273554432t^{15} - 1811939328t^{14} + 132526374912t^{13} + 579112796160t^{12} +
1060210999296t^{11} - 115964116992t^{10} - 5260059869184t^{9} -
13574596460544t^{8} - 20634365067264t^{7} - 25975962206208t^{6} -
35195109507072t^{5} - 46327664738304t^{4} - 45921790328832t^{3} -
29686813949952t^{2} - 11132555231232t - 1855425871872\]
\[B(t) = 54t^{36} + 3888t^{35} + 129600t^{34} + 2643840t^{33} + 36236160t^{32} +
333766656t^{31} + 1728995328t^{30} - 2402058240t^{29} - 143965569024t^{28} -
1516370264064t^{27} - 9489961451520t^{26} - 37456098361344t^{25} -
71662738341888t^{24} + 139076762075136t^{23} + 1527831771217920t^{22} +
5504425254715392t^{21} + 11289182890622976t^{20} + 13175466089250816t^{19} -
105403728714006528t^{17} - 722507704999870464t^{16} - 2818265730414280704t^{15}
- 6257998934908600320t^{14} - 4557267339678056448t^{13} +
18785956879895887872t^{12} + 78551131590689292288t^{11} +
159215133103824568320t^{10} + 203523771649430126592t^{9} +
154581852677027659776t^{8} + 20633523167774638080t^{7} -
118815654219148689408t^{6} - 183490159617956118528t^{5} -
159368317063806320640t^{4} - 93021850303337594880t^{3} -
36479156981701017600t^{2} - 8754997675608244224t - 972777519512027136\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 369951x + 84522549$, with conductor $1470$ |
| Generic density of odd order reductions |
$11/112$ |