| Curve name |
$X_{223c}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{223}$ |
| Curves that $X_{223c}$ minimally covers |
|
| Curves that minimally cover $X_{223c}$ |
|
| Curves that minimally cover $X_{223c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{16} - 1811939328t^{14} - 339738624t^{12} + 28311552t^{10}
+ 4423680t^{8} + 442368t^{6} - 82944t^{4} - 6912t^{2} - 108\]
\[B(t) = 29686813949952t^{24} + 44530220924928t^{22} + 19481971654656t^{20} +
1623497637888t^{18} - 413122166784t^{16} - 65229815808t^{14} + 3170893824t^{12}
- 1019215872t^{10} - 100859904t^{8} + 6193152t^{6} + 1161216t^{4} + 41472t^{2} +
432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 2177x + 36993$, with conductor $3264$ |
| Generic density of odd order reductions |
$109/896$ |