Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 233 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 87\cdot 233^{2} + 158\cdot 233^{3} + 98\cdot 233^{4} + 56\cdot 233^{5} + 134\cdot 233^{6} +O\left(233^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 115\cdot 233 + 170\cdot 233^{2} + 215\cdot 233^{3} + 231\cdot 233^{4} + 134\cdot 233^{5} + 195\cdot 233^{6} +O\left(233^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 82\cdot 233 + 26\cdot 233^{2} + 193\cdot 233^{3} + 190\cdot 233^{4} + 27\cdot 233^{5} + 32\cdot 233^{6} +O\left(233^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 96 + 109\cdot 233 + 184\cdot 233^{2} + 153\cdot 233^{3} + 54\cdot 233^{4} + 191\cdot 233^{5} + 136\cdot 233^{6} +O\left(233^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 137 + 123\cdot 233 + 48\cdot 233^{2} + 79\cdot 233^{3} + 178\cdot 233^{4} + 41\cdot 233^{5} + 96\cdot 233^{6} +O\left(233^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 210 + 150\cdot 233 + 206\cdot 233^{2} + 39\cdot 233^{3} + 42\cdot 233^{4} + 205\cdot 233^{5} + 200\cdot 233^{6} +O\left(233^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 218 + 117\cdot 233 + 62\cdot 233^{2} + 17\cdot 233^{3} + 233^{4} + 98\cdot 233^{5} + 37\cdot 233^{6} +O\left(233^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 226 + 232\cdot 233 + 145\cdot 233^{2} + 74\cdot 233^{3} + 134\cdot 233^{4} + 176\cdot 233^{5} + 98\cdot 233^{6} +O\left(233^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7)(2,8)(3,5)(4,6)$ |
| $(1,8)(4,5)$ |
| $(1,4,8,5)(3,6)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,4,3)(2,5,6,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,4,7)(2,8,6,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,8,5)(3,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,8,4)(3,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
