| Curve name |
$X_{227h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{227}$ |
| Curves that $X_{227h}$ minimally covers |
|
| Curves that minimally cover $X_{227h}$ |
|
| Curves that minimally cover $X_{227h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7077888t^{16} + 212336640t^{14} - 238878720t^{12} + 92897280t^{10} -
30246912t^{8} + 5806080t^{6} - 933120t^{4} + 51840t^{2} - 108\]
\[B(t) = -7247757312t^{24} - 456608710656t^{22} + 1883510931456t^{20} -
1740820709376t^{18} + 942689746944t^{16} - 346023788544t^{14} +
103004504064t^{12} - 21626486784t^{10} + 3682381824t^{8} - 425005056t^{6} +
28740096t^{4} - 435456t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 419199x + 493911585$, with conductor $16320$ |
| Generic density of odd order reductions |
$299/2688$ |