The modular curve $X_{192k}$

Curve name $X_{192k}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 9 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 14 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 15 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $12$ $X_{25}$
$8$ $48$ $X_{192}$
Meaning/Special name
Chosen covering $X_{192}$
Curves that $X_{192k}$ minimally covers
Curves that minimally cover $X_{192k}$
Curves that minimally cover $X_{192k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{30} - 94176t^{28} + 4238784t^{26} - 21261312t^{24} - 1116343296t^{22} + 16016596992t^{20} - 99187310592t^{18} + 426839113728t^{16} - 1586996969472t^{14} + 4100248829952t^{12} - 4572542140416t^{10} - 1393381343232t^{8} + 4444687171584t^{6} - 1580011094016t^{4} - 28991029248t^{2}\] \[B(t) = 432t^{45} - 927936t^{43} - 3773952t^{41} + 4221130752t^{39} - 171411738624t^{37} + 2781887348736t^{35} - 22056893153280t^{33} + 151068346417152t^{31} - 1788549399576576t^{29} + 18664420779491328t^{27} - 120985049792249856t^{25} + 483940199168999424t^{23} - 1194522929887444992t^{21} + 1831474585166413824t^{19} - 2475103787698618368t^{17} + 5782082198773432320t^{15} - 11668081234352799744t^{13} + 11503247055321563136t^{11} - 4532404632994971648t^{9} + 64836001666695168t^{7} + 255069105457987584t^{5} - 1899956092796928t^{3}\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 3388291788x + 75913508557712$, with conductor $141120$
Generic density of odd order reductions $139/1344$

Back to the 2-adic image homepage.