| Curve name |
$X_{193k}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 0 & 9 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{193}$ |
| Curves that $X_{193k}$ minimally covers |
|
| Curves that minimally cover $X_{193k}$ |
|
| Curves that minimally cover $X_{193k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{26} + 3623878656t^{25} + 1811939328t^{24} -
6341787648t^{23} + 1585446912t^{22} + 679477248t^{21} - 339738624t^{20} +
1528823808t^{19} - 2144600064t^{18} + 3312451584t^{17} - 1797783552t^{16} +
771489792t^{15} - 649396224t^{14} - 192872448t^{13} - 112361472t^{12} -
51757056t^{11} - 8377344t^{10} - 1492992t^{9} - 82944t^{8} - 41472t^{7} +
24192t^{6} + 24192t^{5} + 1728t^{4} - 864t^{3} - 108t^{2}\]
\[B(t) = 29686813949952t^{39} - 89060441849856t^{38} + 215229401137152t^{36} -
150289495621632t^{35} - 89060441849856t^{34} + 103903848824832t^{33} -
66795331387392t^{32} - 4174708211712t^{31} + 196211285950464t^{30} -
144723218006016t^{29} - 67839008440320t^{28} + 98366562238464t^{27} -
104715597643776t^{26} + 105933220872192t^{25} - 40964324327424t^{24} +
26420793311232t^{23} - 12532278362112t^{22} - 3133069590528t^{20} -
1651299581952t^{19} - 640067567616t^{18} - 413801644032t^{17} -
102261325824t^{16} - 24015273984t^{15} - 4140564480t^{14} + 2208301056t^{13} +
748486656t^{12} + 3981312t^{11} - 15925248t^{10} - 6193152t^{9} - 1327104t^{8} +
559872t^{7} + 200448t^{6} - 5184t^{4} - 432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 3852075x + 1706652250$, with conductor $25200$ |
| Generic density of odd order reductions |
$139/1344$ |