Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 33\cdot 157 + 69\cdot 157^{2} + 49\cdot 157^{3} + 147\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 69 + 48\cdot 157 + 120\cdot 157^{2} + 125\cdot 157^{3} + 40\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 110 + 124\cdot 157 + 102\cdot 157^{2} + 40\cdot 157^{3} + 13\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 112 + 139\cdot 157 + 148\cdot 157^{2} + 109\cdot 157^{3} + 147\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 114 + 42\cdot 157 + 117\cdot 157^{2} + 60\cdot 157^{3} + 53\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 116 + 32\cdot 157 + 7\cdot 157^{2} + 78\cdot 157^{3} + 72\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 118 + 92\cdot 157 + 132\cdot 157^{2} + 28\cdot 157^{3} + 135\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 133 + 113\cdot 157 + 86\cdot 157^{2} + 134\cdot 157^{3} + 17\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,2,3)(4,6,7,5)$ |
| $(1,2)(3,8)(4,7)(5,6)$ |
| $(1,5,2,6)(3,4,8,7)$ |
| $(3,8)(4,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,2,3)(4,6,7,5)$ | $0$ |
| $4$ | $4$ | $(1,5,2,6)(3,4,8,7)$ | $0$ |
| $2$ | $8$ | $(1,7,8,5,2,4,3,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,4,8,6,2,7,3,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
