| Curve name |
$X_{230g}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{230}$ |
| Curves that $X_{230g}$ minimally covers |
|
| Curves that minimally cover $X_{230g}$ |
|
| Curves that minimally cover $X_{230g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -442368t^{24} + 4423680t^{22} - 16146432t^{20} + 24993792t^{18} -
12358656t^{16} - 3870720t^{14} + 3787776t^{12} - 967680t^{10} - 772416t^{8} +
390528t^{6} - 63072t^{4} + 4320t^{2} - 108\]
\[B(t) = -113246208t^{36} + 1698693120t^{34} - 10446962688t^{32} +
33520877568t^{30} - 58350108672t^{28} + 49347035136t^{26} - 7679508480t^{24} -
15967715328t^{22} + 9807298560t^{20} - 2416214016t^{18} + 2451824640t^{16} -
997982208t^{14} - 119992320t^{12} + 192761856t^{10} - 56982528t^{8} +
8183808t^{6} - 637632t^{4} + 25920t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 12609x + 1199295$, with conductor $9408$ |
| Generic density of odd order reductions |
$271/2688$ |