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:
| $r_{ 1 }$ |
$=$ |
$ 68 a + 61 + \left(110 a + 91\right)\cdot 167 + \left(126 a + 83\right)\cdot 167^{2} + \left(106 a + 4\right)\cdot 167^{3} + \left(62 a + 10\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 130 a + 13 + \left(39 a + 13\right)\cdot 167 + \left(53 a + 134\right)\cdot 167^{2} + 134 a\cdot 167^{3} + \left(113 a + 40\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 99 a + 129 + \left(56 a + 133\right)\cdot 167 + \left(40 a + 99\right)\cdot 167^{2} + \left(60 a + 151\right)\cdot 167^{3} + \left(104 a + 132\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a + 57 + \left(144 a + 35\right)\cdot 167 + \left(117 a + 31\right)\cdot 167^{2} + \left(5 a + 20\right)\cdot 167^{3} + \left(54 a + 125\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 125 a + 99 + \left(22 a + 137\right)\cdot 167 + \left(49 a + 4\right)\cdot 167^{2} + \left(161 a + 75\right)\cdot 167^{3} + \left(112 a + 6\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a + 143 + \left(127 a + 89\right)\cdot 167 + \left(113 a + 147\right)\cdot 167^{2} + \left(32 a + 81\right)\cdot 167^{3} + \left(53 a + 19\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 value |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
