| Curve name |
$X_{229f}$ |
| 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 & 3 \\ 0 & 3 \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 $X_{229f}$ minimally covers |
|
| Curves that minimally cover $X_{229f}$ |
|
| Curves that minimally cover $X_{229f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -442368t^{24} - 4423680t^{22} + 10395648t^{20} + 187342848t^{18} +
485305344t^{16} + 123310080t^{14} - 527053824t^{12} + 30827520t^{10} +
30331584t^{8} + 2927232t^{6} + 40608t^{4} - 4320t^{2} - 108\]
\[B(t) = 113246208t^{36} + 1698693120t^{34} + 24715984896t^{32} +
219018166272t^{30} + 893087907840t^{28} + 1158763511808t^{26} -
2157644611584t^{24} - 6587744256000t^{22} - 1604147576832t^{20} +
4765666738176t^{18} - 401036894208t^{16} - 411734016000t^{14} -
33713197056t^{12} + 4526419968t^{10} + 872156160t^{8} + 53471232t^{6} +
1508544t^{4} + 25920t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 4179711x - 5660476065$, with conductor $55488$ |
| Generic density of odd order reductions |
$109/896$ |