Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 152\cdot 157 + 84\cdot 157^{2} + 91\cdot 157^{3} + 96\cdot 157^{4} + 154\cdot 157^{5} + 30\cdot 157^{6} + 130\cdot 157^{7} +O\left(157^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 24\cdot 157 + 58\cdot 157^{2} + 33\cdot 157^{3} + 103\cdot 157^{4} + 91\cdot 157^{5} + 78\cdot 157^{6} + 23\cdot 157^{7} +O\left(157^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 + 135\cdot 157 + 132\cdot 157^{2} + 64\cdot 157^{3} + 88\cdot 157^{4} + 86\cdot 157^{5} + 96\cdot 157^{6} + 40\cdot 157^{7} +O\left(157^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 76 + 141\cdot 157 + 34\cdot 157^{2} + 154\cdot 157^{3} + 6\cdot 157^{4} + 105\cdot 157^{5} + 103\cdot 157^{6} + 119\cdot 157^{7} +O\left(157^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 81 + 15\cdot 157 + 122\cdot 157^{2} + 2\cdot 157^{3} + 150\cdot 157^{4} + 51\cdot 157^{5} + 53\cdot 157^{6} + 37\cdot 157^{7} +O\left(157^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 112 + 21\cdot 157 + 24\cdot 157^{2} + 92\cdot 157^{3} + 68\cdot 157^{4} + 70\cdot 157^{5} + 60\cdot 157^{6} + 116\cdot 157^{7} +O\left(157^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 131 + 132\cdot 157 + 98\cdot 157^{2} + 123\cdot 157^{3} + 53\cdot 157^{4} + 65\cdot 157^{5} + 78\cdot 157^{6} + 133\cdot 157^{7} +O\left(157^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 154 + 4\cdot 157 + 72\cdot 157^{2} + 65\cdot 157^{3} + 60\cdot 157^{4} + 2\cdot 157^{5} + 126\cdot 157^{6} + 26\cdot 157^{7} +O\left(157^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,5,6)(2,4,3,8)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(1,8)(4,5)$ |
| $(2,7)(3,6)$ |
| $(1,4,8,5)(2,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(2,7)(3,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,5,6)(2,4,3,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,5,7)(2,8,3,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,8,5)(2,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,8,4)(2,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
