| Curve name |
$X_{197d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{197}$ |
| Curves that $X_{197d}$ minimally covers |
|
| Curves that minimally cover $X_{197d}$ |
|
| Curves that minimally cover $X_{197d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 11421t^{24} - 397872t^{23} - 4647888t^{22} - 18249408t^{21} -
56796768t^{20} - 172599552t^{19} - 927058176t^{18} - 3077637120t^{17} -
4617596160t^{16} - 5599715328t^{15} + 26186637312t^{14} + 20061388800t^{13} +
287051268096t^{12} - 80245555200t^{11} + 418986196992t^{10} + 358381780992t^{9}
- 1182104616960t^{8} + 3151500410880t^{7} - 3797230288896t^{6} +
2827871059968t^{5} - 3722232987648t^{4} + 4783972810752t^{3} -
4873663807488t^{2} + 1668796121088t + 191612583936\]
\[B(t) = 3439422t^{36} + 54739152t^{35} + 237311856t^{34} + 1521402048t^{33} +
19285369056t^{32} + 147664373760t^{31} + 710847719424t^{30} +
2112305651712t^{29} + 4322471122944t^{28} + 6642922143744t^{27} +
6274464104448t^{26} + 25067747082240t^{25} + 93932673269760t^{24} +
880070610124800t^{23} + 2508544852623360t^{22} + 7446481927667712t^{21} +
12083511349739520t^{20} + 16713063648460800t^{19} + 28837309694607360t^{18} -
66852254593843200t^{17} + 193336181595832320t^{16} - 476574843370733568t^{15} +
642187482271580160t^{14} - 901192304767795200t^{13} + 384748229712936960t^{12} -
410709968195420160t^{11} + 411203279549104128t^{10} - 1741402182449627136t^{9} +
4532439480212127744t^{8} - 8859652044198248448t^{7} + 11926045731883843584t^{6}
- 9909588376305008640t^{5} + 5176876836675649536t^{4} - 1633593010056855552t^{3}
+ 1019246660473061376t^{2} - 940411470603091968t + 236355280114286592\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 141x + 4718$, with conductor $72$ |
| Generic density of odd order reductions |
$299/2688$ |