Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 139\cdot 157 + 140\cdot 157^{2} + 114\cdot 157^{3} + 31\cdot 157^{4} + 103\cdot 157^{5} + 83\cdot 157^{6} + 143\cdot 157^{7} + 25\cdot 157^{8} + 107\cdot 157^{9} + 43\cdot 157^{10} +O\left(157^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 9\cdot 157 + 16\cdot 157^{2} + 24\cdot 157^{3} + 13\cdot 157^{4} + 52\cdot 157^{5} + 39\cdot 157^{6} + 112\cdot 157^{7} + 117\cdot 157^{8} + 136\cdot 157^{9} + 130\cdot 157^{10} +O\left(157^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 48 + 72\cdot 157 + 24\cdot 157^{2} + 119\cdot 157^{3} + 75\cdot 157^{4} + 62\cdot 157^{5} + 106\cdot 157^{6} + 11\cdot 157^{7} + 103\cdot 157^{8} + 4\cdot 157^{9} + 98\cdot 157^{10} +O\left(157^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 69 + 110\cdot 157 + 22\cdot 157^{2} + 87\cdot 157^{3} + 41\cdot 157^{4} + 41\cdot 157^{5} + 154\cdot 157^{6} + 78\cdot 157^{7} + 38\cdot 157^{8} + 103\cdot 157^{9} + 20\cdot 157^{10} +O\left(157^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 88 + 46\cdot 157 + 134\cdot 157^{2} + 69\cdot 157^{3} + 115\cdot 157^{4} + 115\cdot 157^{5} + 2\cdot 157^{6} + 78\cdot 157^{7} + 118\cdot 157^{8} + 53\cdot 157^{9} + 136\cdot 157^{10} +O\left(157^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 109 + 84\cdot 157 + 132\cdot 157^{2} + 37\cdot 157^{3} + 81\cdot 157^{4} + 94\cdot 157^{5} + 50\cdot 157^{6} + 145\cdot 157^{7} + 53\cdot 157^{8} + 152\cdot 157^{9} + 58\cdot 157^{10} +O\left(157^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 134 + 147\cdot 157 + 140\cdot 157^{2} + 132\cdot 157^{3} + 143\cdot 157^{4} + 104\cdot 157^{5} + 117\cdot 157^{6} + 44\cdot 157^{7} + 39\cdot 157^{8} + 20\cdot 157^{9} + 26\cdot 157^{10} +O\left(157^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 141 + 17\cdot 157 + 16\cdot 157^{2} + 42\cdot 157^{3} + 125\cdot 157^{4} + 53\cdot 157^{5} + 73\cdot 157^{6} + 13\cdot 157^{7} + 131\cdot 157^{8} + 49\cdot 157^{9} + 113\cdot 157^{10} +O\left(157^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4,8,6,7,5)$ |
| $(2,7)(4,5)$ |
| $(1,8)(3,6)$ |
| $(1,8)(2,7)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $8$ | $2$ | $(1,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,7,8,2)$ | $2$ |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,8)(2,7)(3,4,6,5)$ | $-2$ |
| $8$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $0$ |
| $8$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
