| Curve name |
$X_{122k}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{122}$ |
| Curves that $X_{122k}$ minimally covers |
|
| Curves that minimally cover $X_{122k}$ |
|
| Curves that minimally cover $X_{122k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27648t^{12} + 82944t^{10} - 96768t^{8} + 55296t^{6} - 15660t^{4} +
1836t^{2} - 27\]
\[B(t) = 1769472t^{18} - 7962624t^{16} + 15261696t^{14} - 16257024t^{12} +
10502784t^{10} - 4193856t^{8} + 1000944t^{6} - 127656t^{4} + 6156t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 38417x + 2882228$, with conductor $147$ |
| Generic density of odd order reductions |
$25/224$ |