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:
| $r_{ 1 }$ |
$=$ |
$ 89 a + 50 + \left(63 a + 149\right)\cdot 167 + \left(34 a + 77\right)\cdot 167^{2} + \left(a + 66\right)\cdot 167^{3} + \left(3 a + 6\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 107 a + 68 + \left(105 a + 149\right)\cdot 167 + \left(19 a + 125\right)\cdot 167^{2} + \left(68 a + 63\right)\cdot 167^{3} + \left(78 a + 129\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 60 a + 8 + \left(61 a + 148\right)\cdot 167 + \left(147 a + 39\right)\cdot 167^{2} + \left(98 a + 112\right)\cdot 167^{3} + \left(88 a + 139\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 101 + 73\cdot 167 + 118\cdot 167^{2} + 26\cdot 167^{3} + 43\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 135 + 23\cdot 167 + 90\cdot 167^{2} + 31\cdot 167^{3} + 7\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 78 a + 139 + \left(103 a + 123\right)\cdot 167 + \left(132 a + 48\right)\cdot 167^{2} + \left(165 a + 33\right)\cdot 167^{3} + \left(163 a + 8\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$16$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $15$ |
$2$ |
$(1,2)$ |
$0$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
