| Curve name |
$X_{217b}$ |
| 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 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{217}$ |
| Curves that $X_{217b}$ minimally covers |
|
| Curves that minimally cover $X_{217b}$ |
|
| Curves that minimally cover $X_{217b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} + 13392t^{14} - 390096t^{12} + 3471552t^{10} - 11089440t^{8}
+ 13886208t^{6} - 6241536t^{4} + 857088t^{2} - 6912\]
\[B(t) = -54t^{24} - 53136t^{22} + 4672080t^{20} - 116036928t^{18} +
1270274400t^{16} - 6784528896t^{14} + 18672256512t^{12} - 27138115584t^{10} +
20324390400t^{8} - 7426363392t^{6} + 1196052480t^{4} - 54411264t^{2} - 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 6176x - 69388$, with conductor $336$ |
| Generic density of odd order reductions |
$53/896$ |