Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 37\cdot 61 + 2\cdot 61^{2} + 16\cdot 61^{3} + 38\cdot 61^{4} + 9\cdot 61^{5} + 53\cdot 61^{6} + 6\cdot 61^{7} + 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 11\cdot 61 + 31\cdot 61^{2} + 33\cdot 61^{3} + 49\cdot 61^{4} + 7\cdot 61^{5} + 2\cdot 61^{6} + 44\cdot 61^{7} + 36\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 43\cdot 61 + 12\cdot 61^{2} + 17\cdot 61^{3} + 36\cdot 61^{4} + 20\cdot 61^{5} + 38\cdot 61^{6} + 55\cdot 61^{7} + 43\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 2\cdot 61 + 15\cdot 61^{2} + 22\cdot 61^{3} + 8\cdot 61^{4} + 30\cdot 61^{5} + 58\cdot 61^{6} + 14\cdot 61^{7} + 55\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 26\cdot 61 + 60\cdot 61^{2} + 32\cdot 61^{3} + 50\cdot 61^{4} + 53\cdot 61^{5} + 30\cdot 61^{6} + 46\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
