| Curve name |
$X_{200e}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{200}$ |
| Curves that $X_{200e}$ minimally covers |
|
| Curves that minimally cover $X_{200e}$ |
|
| Curves that minimally cover $X_{200e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -110592t^{24} + 1105920t^{22} - 10672128t^{20} + 59332608t^{18} -
157365504t^{16} + 208051200t^{14} - 150011136t^{12} + 52012800t^{10} -
9835344t^{8} + 927072t^{6} - 41688t^{4} + 1080t^{2} - 27\]
\[B(t) = 14155776t^{36} - 212336640t^{34} - 477757440t^{32} + 18997051392t^{30}
- 126478319616t^{28} + 456237121536t^{26} - 1060075634688t^{24} +
1623202136064t^{22} - 1586118537216t^{20} + 982579286016t^{18} -
396529634304t^{16} + 101450133504t^{14} - 16563681792t^{12} + 1782176256t^{10} -
123513984t^{8} + 4637952t^{6} - 29160t^{4} - 3240t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 5097x - 49995$, with conductor $294$ |
| Generic density of odd order reductions |
$81/896$ |