| Curve name |
$X_{122p}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 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],
\left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{122}$ |
| Curves that $X_{122p}$ minimally covers |
|
| Curves that minimally cover $X_{122p}$ |
|
| Curves that minimally cover $X_{122p}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -110592t^{16} + 552960t^{14} - 1161216t^{12} + 1327104t^{10} -
892080t^{8} + 353808t^{6} - 77436t^{4} + 7560t^{2} - 108\]
\[B(t) = -14155776t^{24} + 106168320t^{22} - 355663872t^{20} + 701595648t^{18} -
904172544t^{16} + 797879808t^{14} - 490783104t^{12} + 209718720t^{10} -
60686496t^{8} + 11218608t^{6} - 1167696t^{4} + 47952t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 22128204x - 40065204368$, with conductor $28224$ |
| Generic density of odd order reductions |
$635/5376$ |