| Curve name |
$X_{185d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{185}$ |
| Curves that $X_{185d}$ minimally covers |
|
| Curves that minimally cover $X_{185d}$ |
|
| Curves that minimally cover $X_{185d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -110592t^{24} - 1603584t^{20} - 103680t^{16} + 255744t^{12} - 6480t^{8}
- 6264t^{4} - 27\]
\[B(t) = 14155776t^{36} - 456523776t^{32} - 583925760t^{28} + 216760320t^{24} +
69341184t^{20} - 17335296t^{16} - 3386880t^{12} + 570240t^{8} + 27864t^{4} -
54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 1215x + 16600$, with conductor $45$ |
| Generic density of odd order reductions |
$299/2688$ |