| Curve name |
$X_{123e}$ |
| Index |
$48$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 31 & 31 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 11 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 21 & 0 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{123}$ |
| Curves that $X_{123e}$ minimally covers |
|
| Curves that minimally cover $X_{123e}$ |
|
| Curves that minimally cover $X_{123e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -189t^{20} - 5076t^{19} - 60696t^{18} - 438696t^{17} - 2203524t^{16} -
8351424t^{15} - 25090560t^{14} - 61160832t^{13} - 121204512t^{12} -
192516480t^{11} - 238028544t^{10} - 216822528t^{9} - 126831744t^{8} -
20487168t^{7} + 40144896t^{6} + 41306112t^{5} + 16899840t^{4} + 248832t^{3} -
2930688t^{2} - 1271808t - 193536\]
\[B(t) = 918t^{30} + 35316t^{29} + 612684t^{28} + 6326208t^{27} + 42797160t^{26}
+ 191791152t^{25} + 494829648t^{24} - 102269952t^{23} - 7988961312t^{22} -
44932459968t^{21} - 160949726784t^{20} - 432735851520t^{19} - 920784143232t^{18}
- 1583438637312t^{17} - 2211355365120t^{16} - 2484482015232t^{15} -
2174846049792t^{14} - 1352611685376t^{13} - 387801603072t^{12} +
295069630464t^{11} + 504433391616t^{10} + 359680020480t^{9} + 123138994176t^{8}
- 21467234304t^{7} - 49863942144t^{6} - 26113425408t^{5} - 4654153728t^{4} +
1833172992t^{3} + 1324449792t^{2} + 323813376t + 30081024\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 12871434x - 17769846862$, with conductor $21632$ |
| Generic density of odd order reductions |
$2722915/11010048$ |