| Curve name |
$X_{61e}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 6 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 10 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 12 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{61}$ |
| Curves that $X_{61e}$ minimally covers |
|
| Curves that minimally cover $X_{61e}$ |
|
| Curves that minimally cover $X_{61e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{18} - 324t^{14} - 432t^{10} - 324t^{6} - 108t^{2}\]
\[B(t) = 432t^{27} + 1944t^{23} + 2592t^{19} - 2592t^{11} - 1944t^{7} -
432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 14312853x - 18640762520$, with conductor
$84681$ |
| Generic density of odd order reductions |
$9249/57344$ |