Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 233 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 178\cdot 233 + 66\cdot 233^{2} + 115\cdot 233^{3} + 67\cdot 233^{4} + 151\cdot 233^{5} + 148\cdot 233^{6} + 133\cdot 233^{7} +O\left(233^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 84\cdot 233 + 115\cdot 233^{2} + 175\cdot 233^{3} + 157\cdot 233^{4} + 11\cdot 233^{5} + 198\cdot 233^{6} + 182\cdot 233^{7} +O\left(233^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 102\cdot 233 + 190\cdot 233^{2} + 119\cdot 233^{3} + 78\cdot 233^{4} + 30\cdot 233^{5} + 115\cdot 233^{6} + 218\cdot 233^{7} +O\left(233^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 103 + 72\cdot 233 + 156\cdot 233^{2} + 62\cdot 233^{3} + 88\cdot 233^{4} + 142\cdot 233^{5} + 76\cdot 233^{6} + 58\cdot 233^{7} +O\left(233^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 107 + 13\cdot 233 + 134\cdot 233^{2} + 167\cdot 233^{3} + 134\cdot 233^{4} + 42\cdot 233^{5} + 222\cdot 233^{6} + 51\cdot 233^{7} +O\left(233^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 144 + 125\cdot 233 + 38\cdot 233^{2} + 38\cdot 233^{3} + 137\cdot 233^{4} + 18\cdot 233^{5} + 75\cdot 233^{6} + 130\cdot 233^{7} +O\left(233^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 148 + 9\cdot 233 + 159\cdot 233^{2} + 121\cdot 233^{3} + 80\cdot 233^{4} + 199\cdot 233^{6} + 225\cdot 233^{7} +O\left(233^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 149 + 113\cdot 233 + 71\cdot 233^{2} + 131\cdot 233^{3} + 187\cdot 233^{4} + 68\cdot 233^{5} + 130\cdot 233^{6} + 163\cdot 233^{7} +O\left(233^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,5)(6,7)$ |
| $(1,5)(2,6)$ |
| $(1,7)(2,3)(4,5)(6,8)$ |
| $(1,2)(5,6)$ |
| $(4,8)(5,6)$ |
| $(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,2)(3,7)(4,8)(5,6)$ |
$-4$ |
| $2$ |
$2$ |
$(3,7)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,4)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,5)(3,4)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,8)(3,5)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,3)(4,5)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(4,8)(5,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,6)$ |
$2$ |
| $4$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-2$ |
| $4$ |
$4$ |
$(1,5,2,6)(3,4,7,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,2,8)(3,6,7,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,2,8)(3,5,7,6)$ |
$0$ |
| $8$ |
$4$ |
$(1,4,5,3)(2,8,6,7)$ |
$0$ |
| $8$ |
$4$ |
$(1,4,6,3)(2,8,5,7)$ |
$0$ |
| $8$ |
$4$ |
$(1,2)(3,4,7,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
