| Curve name |
$X_{193g}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{193}$ |
| Curves that $X_{193g}$ minimally covers |
|
| Curves that minimally cover $X_{193g}$ |
|
| Curves that minimally cover $X_{193g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{24} + 2264924160t^{22} - 4133486592t^{20} +
3199205376t^{18} - 1215627264t^{16} + 1150156800t^{14} - 947994624t^{12} +
71884800t^{10} - 4748544t^{8} + 781056t^{6} - 63072t^{4} + 2160t^{2} - 27\]
\[B(t) = 3710851743744t^{36} - 27831388078080t^{34} + 85581518340096t^{32} -
137301514518528t^{30} + 112195283189760t^{28} - 10349797441536t^{26} -
78714268286976t^{24} + 79884781092864t^{22} - 28937237299200t^{20} +
3151132360704t^{18} - 1808577331200t^{16} + 312049926144t^{14} -
19217350656t^{12} - 157925376t^{10} + 106997760t^{8} - 8183808t^{6} +
318816t^{4} - 6480t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 52431x - 2731947$, with conductor $1470$ |
| Generic density of odd order reductions |
$11/112$ |