| Curve name |
$X_{192e}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 6 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192e}$ minimally covers |
|
| Curves that minimally cover $X_{192e}$ |
|
| Curves that minimally cover $X_{192e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{22} - 24840t^{20} - 116208t^{18} + 4188672t^{16} - 26445312t^{14}
+ 105172992t^{12} - 423124992t^{10} + 1072300032t^{8} - 475987968t^{6} -
1627914240t^{4} - 28311552t^{2}\]
\[B(t) = 54t^{33} - 112104t^{31} - 8639136t^{29} + 105591168t^{27} -
290345472t^{25} + 3155687424t^{23} - 67851067392t^{21} + 551817805824t^{19} -
2207271223296t^{17} + 4342468313088t^{15} - 3231423922176t^{13} +
4757020213248t^{11} - 27680091144192t^{9} + 36235162681344t^{7} +
7523172089856t^{5} - 57982058496t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 1080450x + 432540000$, with conductor $630$ |
| Generic density of odd order reductions |
$271/2688$ |