| Curve name |
$X_{190c}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{190}$ |
| Curves that $X_{190c}$ minimally covers |
|
| Curves that minimally cover $X_{190c}$ |
|
| Curves that minimally cover $X_{190c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -351t^{16} - 10368t^{15} - 157248t^{14} - 1658880t^{13} -
13706496t^{12} - 91570176t^{11} - 488927232t^{10} - 2043740160t^{9} -
6594600960t^{8} - 16349921280t^{7} - 31291342848t^{6} - 46883930112t^{5} -
56141807616t^{4} - 54358179840t^{3} - 41221619712t^{2} - 21743271936t -
5888802816\]
\[B(t) = 1890t^{24} + 62208t^{23} + 710208t^{22} - 2239488t^{21} -
190003968t^{20} - 3218890752t^{19} - 34154459136t^{18} - 266174595072t^{17} -
1624896184320t^{16} - 8064035979264t^{15} - 33396264271872t^{14} -
117705164193792t^{13} - 357696245071872t^{12} - 941641313550336t^{11} -
2137360913399808t^{10} - 4128786421383168t^{9} - 6655574770974720t^{8} -
8722009131319296t^{7} - 8953386535747584t^{6} - 6750503178338304t^{5} -
3187737611993088t^{4} - 300578991243264t^{3} + 762580033339392t^{2} +
534362651099136t + 129879811031040\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 4x + 4$, with conductor $24$ |
| Generic density of odd order reductions |
$215/2688$ |