| Curve name |
$X_{227k}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{227}$ |
| Curves that $X_{227k}$ minimally covers |
|
| Curves that minimally cover $X_{227k}$ |
|
| Curves that minimally cover $X_{227k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1769472t^{16} + 53084160t^{14} - 59719680t^{12} + 23224320t^{10} -
7561728t^{8} + 1451520t^{6} - 233280t^{4} + 12960t^{2} - 27\]
\[B(t) = 905969664t^{24} + 57076088832t^{22} - 235438866432t^{20} +
217602588672t^{18} - 117836218368t^{16} + 43252973568t^{14} - 12875563008t^{12}
+ 2703310848t^{10} - 460297728t^{8} + 53125632t^{6} - 3592512t^{4} + 54432t^{2}
+ 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 + 6550x - 962215$, with conductor $510$ |
| Generic density of odd order reductions |
$19/336$ |