Curve name | $X_{194d}$ | ||||||||||||
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} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$ | ||||||||||||
Images in lower levels |
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Meaning/Special name | |||||||||||||
Chosen covering | $X_{194}$ | ||||||||||||
Curves that $X_{194d}$ minimally covers | |||||||||||||
Curves that minimally cover $X_{194d}$ | |||||||||||||
Curves that minimally cover $X_{194d}$ and have infinitely many rational points. | |||||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -28991029248t^{30} - 159450660864t^{28} - 338832654336t^{26} - 346080411648t^{24} - 196708663296t^{22} - 125306929152t^{20} - 102339182592t^{18} - 39537082368t^{16} - 6396198912t^{14} - 489480192t^{12} - 48024576t^{10} - 5280768t^{8} - 323136t^{6} - 9504t^{4} - 108t^{2}\] \[B(t) = 1899956092796928t^{45} + 15674637765574656t^{43} + 54861232179511296t^{41} + 105863178545528832t^{39} + 118606243433545728t^{37} + 62247682575433728t^{35} - 24458223843016704t^{33} - 69236144121839616t^{31} - 52965392812867584t^{29} - 21023932022784000t^{27} - 5553079449550848t^{25} - 1388269862387712t^{23} - 328498937856000t^{21} - 51724016418816t^{19} - 4225838874624t^{17} - 93300719616t^{15} + 14841004032t^{13} + 1767370752t^{11} + 98592768t^{9} + 3193344t^{7} + 57024t^{5} + 432t^{3}\] | ||||||||||||
Info about rational points | |||||||||||||
Comments on finding rational points | None | ||||||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 + xy + y = x^3 - x^2 - 469481855x + 3666831114147$, with conductor $130050$ | ||||||||||||
Generic density of odd order reductions | $299/2688$ |