| Curve name |
$X_{227a}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 16 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{227}$ |
| Curves that $X_{227a}$ minimally covers |
|
| Curves that minimally cover $X_{227a}$ |
|
| Curves that minimally cover $X_{227a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{24} + 13589544960t^{22} - 15231614976t^{20} +
4246732800t^{18} - 26542080t^{16} - 318504960t^{14} + 122535936t^{12} -
19906560t^{10} - 103680t^{8} + 1036800t^{6} - 232416t^{4} + 12960t^{2} - 27\]
\[B(t) = 3710851743744t^{36} + 233783659855872t^{34} - 965053381607424t^{32} +
847465766977536t^{30} - 301796614471680t^{28} + 12785043898368t^{26} +
26457938067456t^{24} - 11757674299392t^{22} + 2582353281024t^{20} -
161397080064t^{16} + 45928415232t^{14} - 6459457536t^{12} - 195084288t^{10} +
287815680t^{8} - 50512896t^{6} + 3595104t^{4} - 54432t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + 583749x + 55853398$, with conductor $6150$ |
| Generic density of odd order reductions |
$73/672$ |