| Curve name |
$X_{203b}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{203}$ |
| Curves that $X_{203b}$ minimally covers |
|
| Curves that minimally cover $X_{203b}$ |
|
| Curves that minimally cover $X_{203b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{16} - 1811939328t^{14} - 7134511104t^{12} -
3369074688t^{10} - 738754560t^{8} - 52641792t^{6} - 1741824t^{4} - 6912t^{2} -
108\]
\[B(t) = 29686813949952t^{24} + 44530220924928t^{22} - 214301688201216t^{20} -
232160162217984t^{18} - 164792258002944t^{16} - 56684709937152t^{14} -
9357307674624t^{12} - 885698592768t^{10} - 40232484864t^{8} - 885620736t^{6} -
12773376t^{4} + 41472t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 110977x - 14192255$, with conductor $3264$ |
| Generic density of odd order reductions |
$109/896$ |