Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $ x^{2} + 166 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 3 + \left(150 a + 82\right)\cdot 167 + \left(89 a + 154\right)\cdot 167^{2} + \left(128 a + 73\right)\cdot 167^{3} + \left(142 a + 128\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 85 a + 8 + \left(67 a + 69\right)\cdot 167 + \left(33 a + 122\right)\cdot 167^{2} + \left(97 a + 61\right)\cdot 167^{3} + \left(128 a + 31\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 146 a + 24 + \left(16 a + 44\right)\cdot 167 + \left(77 a + 94\right)\cdot 167^{2} + \left(38 a + 112\right)\cdot 167^{3} + \left(24 a + 142\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a + 5 + \left(60 a + 29\right)\cdot 167 + \left(104 a + 82\right)\cdot 167^{2} + \left(106 a + 62\right)\cdot 167^{3} + \left(58 a + 8\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 82 a + 93 + \left(99 a + 51\right)\cdot 167 + \left(133 a + 88\right)\cdot 167^{2} + \left(69 a + 125\right)\cdot 167^{3} + \left(38 a + 62\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 136 a + 36 + \left(106 a + 58\right)\cdot 167 + \left(62 a + 126\right)\cdot 167^{2} + \left(60 a + 64\right)\cdot 167^{3} + \left(108 a + 127\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $15$ |
$2$ |
$(1,2)$ |
$-1$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
