| Curve name |
$X_{205g}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{205}$ |
| Curves that $X_{205g}$ minimally covers |
|
| Curves that minimally cover $X_{205g}$ |
|
| Curves that minimally cover $X_{205g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} + 1674t^{20} - 6885t^{16} + 10476t^{12} - 6885t^{8} +
1674t^{4} - 27\]
\[B(t) = 54t^{36} + 6642t^{32} - 76464t^{28} + 284256t^{24} - 517428t^{20} +
517428t^{16} - 284256t^{12} + 76464t^{8} - 6642t^{4} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 391751x - 1532429602$, with conductor $6150$ |
| Generic density of odd order reductions |
$73/672$ |