| Curve name |
$X_{230c}$ |
| 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 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{230}$ |
| Curves that $X_{230c}$ minimally covers |
|
| Curves that minimally cover $X_{230c}$ |
|
| Curves that minimally cover $X_{230c}$ 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 |
$109/896$ |