| Curve name |
$X_{194b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 8 & 5 \end{matrix}\right],
\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]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{194}$ |
| Curves that $X_{194b}$ minimally covers |
|
| Curves that minimally cover $X_{194b}$ |
|
| Curves that minimally cover $X_{194b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -113246208t^{22} - 283115520t^{20} - 205258752t^{18} - 42467328t^{16} -
99975168t^{14} - 49102848t^{12} - 6248448t^{10} - 165888t^{8} - 50112t^{6} -
4320t^{4} - 108t^{2}\]
\[B(t) = 463856467968t^{33} + 1739461754880t^{31} + 2348273369088t^{29} +
1384321646592t^{27} - 614247432192t^{25} - 1651129712640t^{23} -
779473649664t^{21} - 164348559360t^{19} - 41087139840t^{17} - 12179275776t^{15}
- 1612431360t^{13} - 37490688t^{11} + 5280768t^{9} + 559872t^{7} + 25920t^{5} +
432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 9872033x - 3668315937$, with conductor $196800$ |
| Generic density of odd order reductions |
$299/2688$ |