Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 23\cdot 47 + 24\cdot 47^{2} + 10\cdot 47^{3} + 45\cdot 47^{4} + 30\cdot 47^{5} + 5\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 8\cdot 47 + 45\cdot 47^{2} + 23\cdot 47^{3} + 2\cdot 47^{4} + 9\cdot 47^{5} + 23\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 30\cdot 47 + 19\cdot 47^{2} + 2\cdot 47^{3} + 12\cdot 47^{4} + 23\cdot 47^{5} + 17\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 8\cdot 47 + 27\cdot 47^{2} + 22\cdot 47^{3} + 34\cdot 47^{4} + 20\cdot 47^{5} + 14\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 38\cdot 47 + 19\cdot 47^{2} + 24\cdot 47^{3} + 12\cdot 47^{4} + 26\cdot 47^{5} + 32\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 + 16\cdot 47 + 27\cdot 47^{2} + 44\cdot 47^{3} + 34\cdot 47^{4} + 23\cdot 47^{5} + 29\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 30 + 38\cdot 47 + 47^{2} + 23\cdot 47^{3} + 44\cdot 47^{4} + 37\cdot 47^{5} + 23\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 46 + 23\cdot 47 + 22\cdot 47^{2} + 36\cdot 47^{3} + 47^{4} + 16\cdot 47^{5} + 41\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,8,3)(2,4,7,5)$ |
| $(1,5,3,7,8,4,6,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-1$ |
$-1$ |
$-1$ |
$-1$ |
| $1$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
| $1$ |
$4$ |
$(1,6,8,3)(2,4,7,5)$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
| $1$ |
$8$ |
$(1,5,3,7,8,4,6,2)$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,7,6,5,8,2,3,4)$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
| $1$ |
$8$ |
$(1,4,3,2,8,5,6,7)$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,2,6,4,8,7,3,5)$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
