| Curve name |
$X_{212k}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{212}$ |
| Curves that $X_{212k}$ minimally covers |
|
| Curves that minimally cover $X_{212k}$ |
|
| Curves that minimally cover $X_{212k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{16} + 27179089920t^{14} - 15288238080t^{12} +
2972712960t^{10} - 483950592t^{8} + 46448640t^{6} - 3732480t^{4} + 103680t^{2} -
108\]
\[B(t) = 29686813949952t^{24} + 935134639423488t^{22} - 1928715193810944t^{20} +
891300203200512t^{18} - 241328575217664t^{16} + 44291044933632t^{14} -
6592288260096t^{12} + 692047577088t^{10} - 58918109184t^{8} + 3400040448t^{6} -
114960384t^{4} + 870912t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 7041x - 429345$, with conductor $960$ |
| Generic density of odd order reductions |
$299/2688$ |