| Curve name |
$X_{108c}$ |
| Index |
$48$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 21 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 10 & 21 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{108}$ |
| Curves that $X_{108c}$ minimally covers |
|
| Curves that minimally cover $X_{108c}$ |
|
| Curves that minimally cover $X_{108c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 54t^{20} - 216t^{19} - 6912t^{18} - 24624t^{17} + 87480t^{16} +
808704t^{15} + 1689984t^{14} - 2778624t^{13} - 22814784t^{12} - 53450496t^{11} -
47402496t^{10} + 61309440t^{9} + 266727168t^{8} + 446570496t^{7} +
468633600t^{6} + 334430208t^{5} + 163524096t^{4} + 53250048t^{3} + 10838016t^{2}
+ 1216512t + 55296\]
\[B(t) = -540t^{30} - 8424t^{29} - 27864t^{28} + 229824t^{27} + 1932336t^{26} +
1796256t^{25} - 30878496t^{24} - 121678848t^{23} + 70279488t^{22} +
1630088064t^{21} + 3708270720t^{20} - 5403718656t^{19} - 49006487808t^{18} -
109890805248t^{17} - 9524169216t^{16} + 648125743104t^{15} + 2228816342016t^{14}
+ 4541579642880t^{13} + 6586419640320t^{12} + 7132383756288t^{11} +
5773489311744t^{10} + 3339683094528t^{9} + 1154721816576t^{8} - 26435911680t^{7}
- 319026511872t^{6} - 220668198912t^{5} - 88345313280t^{4} - 23137615872t^{3} -
3922919424t^{2} - 392822784t - 17694720\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 3217647x + 2223146015$, with conductor $21632$ |
| Generic density of odd order reductions |
$2722915/11010048$ |