| Curve name |
$X_{202d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{202}$ |
| Curves that $X_{202d}$ minimally covers |
|
| Curves that minimally cover $X_{202d}$ |
|
| Curves that minimally cover $X_{202d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -110592t^{24} - 12165120t^{22} + 55683072t^{20} + 16201728t^{18} -
276804864t^{16} + 219663360t^{14} - 98585856t^{12} + 54915840t^{10} -
17300304t^{8} + 253152t^{6} + 217512t^{4} - 11880t^{2} - 27\]
\[B(t) = -14155776t^{36} + 3779592192t^{34} + 7612268544t^{32} -
261570428928t^{30} + 973701513216t^{28} - 1124205723648t^{26} +
307830620160t^{24} - 933830000640t^{22} + 1721116864512t^{20} -
1236216729600t^{18} + 430279216128t^{16} - 58364375040t^{14} + 4809853440t^{12}
- 4391428608t^{10} + 950880384t^{8} - 63859968t^{6} + 464616t^{4} + 57672t^{2} -
54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 716592x - 231003072$, with conductor $2352$ |
| Generic density of odd order reductions |
$299/2688$ |