| Curve name |
$X_{199b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{199}$ |
| Curves that $X_{199b}$ minimally covers |
|
| Curves that minimally cover $X_{199b}$ |
|
| Curves that minimally cover $X_{199b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} + 11880t^{22} + 217512t^{20} - 253152t^{18} - 17300304t^{16}
- 54915840t^{14} - 98585856t^{12} - 219663360t^{10} - 276804864t^{8} -
16201728t^{6} + 55683072t^{4} + 12165120t^{2} - 110592\]
\[B(t) = 54t^{36} + 57672t^{34} - 464616t^{32} - 63859968t^{30} -
950880384t^{28} - 4391428608t^{26} - 4809853440t^{24} - 58364375040t^{22} -
430279216128t^{20} - 1236216729600t^{18} - 1721116864512t^{16} -
933830000640t^{14} - 307830620160t^{12} - 1124205723648t^{10} -
973701513216t^{8} - 261570428928t^{6} - 7612268544t^{4} + 3779592192t^{2} +
14155776\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 475122x - 151542927$, with conductor $1734$ |
| Generic density of odd order reductions |
$299/2688$ |