| Curve name |
$X_{36r}$ |
| Index |
$24$ |
| Level |
$16$ |
| 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 \\ 8 & 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 $X_{36r}$ minimally covers |
|
| Curves that minimally cover $X_{36r}$ |
|
| Curves that minimally cover $X_{36r}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{6} + 864t^{5} - 13824t^{3} + 25920t^{2} + 13824t - 27648\]
\[B(t) = 432t^{9} - 5184t^{8} + 10368t^{7} + 96768t^{6} - 445824t^{5} +
41472t^{4} + 2515968t^{3} - 3649536t^{2} + 1327104t - 1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 40671x + 3167194$, with conductor $693$ |
| Generic density of odd order reductions |
$19/168$ |