| Curve name |
$X_{229g}$ |
| 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} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{229}$ |
| Curves that $X_{229g}$ minimally covers |
|
| Curves that minimally cover $X_{229g}$ |
|
| Curves that minimally cover $X_{229g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -6912t^{16} - 27648t^{14} + 393984t^{12} + 836352t^{10} - 1136160t^{8}
+ 209088t^{6} + 24624t^{4} - 432t^{2} - 27\]
\[B(t) = 221184t^{24} + 1327104t^{22} + 30191616t^{20} + 112250880t^{18} -
147101184t^{16} - 488208384t^{14} + 515676672t^{12} - 122052096t^{10} -
9193824t^{8} + 1753920t^{6} + 117936t^{4} + 1296t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + 226x - 2232$, with conductor $102$ |
| Generic density of odd order reductions |
$53/896$ |