The modular curve $X_{197c}$

Curve name	$X_{197c}$
Index	$96$
Level	$16$
Genus	$0$
Does the subgroup contain $-I$?	No
Generating matrices	$\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $48$ $X_{197}$
Meaning/Special name
Chosen covering	$X_{197}$
Curves that $X_{197c}$ minimally covers
Curves that minimally cover $X_{197c}$
Curves that minimally cover $X_{197c}$ 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 $144$
Generic density of odd order reductions	$299/2688$

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