| Curve name |
$X_{10c}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 2 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{10}$ |
| Curves that $X_{10c}$ minimally covers |
|
| Curves that minimally cover $X_{10c}$ |
|
| Curves that minimally cover $X_{10c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 324t^{8} + 540t^{6} + 108t^{4} - 108t^{2}\]
\[B(t) = 3888t^{11} + 12096t^{9} + 12960t^{7} + 5184t^{5} + 432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 92x + 312$, with conductor $100$ |
| Generic density of odd order reductions |
$2659/10752$ |