| Curve name |
$X_{193b}$ |
| 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 & 14 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 10 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{193}$ |
| Curves that $X_{193b}$ minimally covers |
|
| Curves that minimally cover $X_{193b}$ |
|
| Curves that minimally cover $X_{193b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -28311552t^{22} + 70778880t^{20} - 51314688t^{18} + 10616832t^{16} -
24993792t^{14} + 12275712t^{12} - 1562112t^{10} + 41472t^{8} - 12528t^{6} +
1080t^{4} - 27t^{2}\]
\[B(t) = 57982058496t^{33} - 217432719360t^{31} + 293534171136t^{29} -
173040205824t^{27} - 76780929024t^{25} + 206391214080t^{23} - 97434206208t^{21}
+ 20543569920t^{19} - 5135892480t^{17} + 1522409472t^{15} - 201553920t^{13} +
4686336t^{11} + 660096t^{9} - 69984t^{7} + 3240t^{5} - 54t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 9630x - 210924$, with conductor $630$ |
| Generic density of odd order reductions |
$271/2688$ |