Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 44\cdot 157 + 49\cdot 157^{2} + 112\cdot 157^{3} + 101\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 147\cdot 157 + 143\cdot 157^{2} + 38\cdot 157^{3} + 142\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 + 82\cdot 157 + 145\cdot 157^{2} + 104\cdot 157^{3} + 51\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 8\cdot 157 + 11\cdot 157^{2} + 19\cdot 157^{3} + 65\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 65 + 83\cdot 157 + 157^{2} + 81\cdot 157^{3} + 103\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 66 + 104\cdot 157 + 28\cdot 157^{2} + 136\cdot 157^{3} + 63\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 96 + 58\cdot 157 + 55\cdot 157^{2} + 89\cdot 157^{3} +O\left(157^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 143 + 99\cdot 157 + 35\cdot 157^{2} + 46\cdot 157^{3} + 99\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,8,3)(2,5,4,7)$ |
| $(1,8)(2,4)(3,6)(5,7)$ |
| $(1,8)(3,6)$ |
| $(1,5,8,7)(2,6,4,3)$ |
| $(1,4,3,7,8,2,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,4)(3,6)(5,7)$ |
$-4$ |
| $2$ |
$2$ |
$(1,8)(3,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(3,8)(5,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,5)(4,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,7)(4,8)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,5,4,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,8,3)(2,5,4,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,8,5)(2,3,4,6)$ |
$0$ |
| $4$ |
$8$ |
$(1,4,3,7,8,2,6,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,3,4,8,7,6,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
