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:
| $r_{ 1 }$ |
$=$ |
$ 87 + 12\cdot 167 + 50\cdot 167^{2} + 43\cdot 167^{3} + 126\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 164 a + 153 + \left(76 a + 64\right)\cdot 167 + \left(157 a + 3\right)\cdot 167^{2} + \left(102 a + 165\right)\cdot 167^{3} + \left(12 a + 120\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 a + 23 + \left(21 a + 63\right)\cdot 167 + \left(41 a + 142\right)\cdot 167^{2} + \left(47 a + 1\right)\cdot 167^{3} + \left(16 a + 42\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 + 21\cdot 167 + 59\cdot 167^{2} + 5\cdot 167^{3} + 3\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 150 + \left(90 a + 144\right)\cdot 167 + \left(9 a + 83\right)\cdot 167^{2} + \left(64 a + 110\right)\cdot 167^{3} + \left(154 a + 30\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 110 a + 80 + \left(145 a + 27\right)\cdot 167 + \left(125 a + 162\right)\cdot 167^{2} + \left(119 a + 7\right)\cdot 167^{3} + \left(150 a + 11\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 |
| $1$ | $1$ | $()$ | $16$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,2)$ | $0$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)$ | $-2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
