| Curve name |
$X_{229c}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{229}$ |
| Curves that $X_{229c}$ minimally covers |
|
| Curves that minimally cover $X_{229c}$ |
|
| Curves that minimally cover $X_{229c}$ 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 |
$y^2 = x^3 - x^2 + 3616x + 142848$, with conductor $816$ |
| Generic density of odd order reductions |
$215/2688$ |