| Curve name |
$X_{36q}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{36}$ |
| Curves that $X_{36q}$ minimally covers |
|
| Curves that minimally cover $X_{36q}$ |
|
| Curves that minimally cover $X_{36q}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{8} + 864t^{7} - 13824t^{5} + 25920t^{4} + 13824t^{3} -
27648t^{2}\]
\[B(t) = 432t^{12} - 5184t^{11} + 10368t^{10} + 96768t^{9} - 445824t^{8} +
41472t^{7} + 2515968t^{6} - 3649536t^{5} + 1327104t^{4} - 1769472t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 1539765x + 735795481$, with conductor $990$ |
| Generic density of odd order reductions |
$83/672$ |