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:
| $r_{ 1 }$ |
$=$ |
$ 144 + 16\cdot 167 + 146\cdot 167^{2} + 130\cdot 167^{3} + 61\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 + 84\cdot 167 + 31\cdot 167^{2} + 68\cdot 167^{3} + 89\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 83 a + 16 + \left(72 a + 80\right)\cdot 167 + \left(71 a + 93\right)\cdot 167^{2} + \left(14 a + 104\right)\cdot 167^{3} + \left(48 a + 10\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 41 + \left(126 a + 134\right)\cdot 167 + \left(62 a + 7\right)\cdot 167^{2} + \left(129 a + 10\right)\cdot 167^{3} + \left(41 a + 148\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 84 a + 99 + \left(94 a + 69\right)\cdot 167 + \left(95 a + 92\right)\cdot 167^{2} + \left(152 a + 47\right)\cdot 167^{3} + \left(118 a + 44\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 132 a + 76 + \left(40 a + 58\right)\cdot 167 + \left(104 a + 111\right)\cdot 167^{2} + \left(37 a + 76\right)\cdot 167^{3} + \left(125 a + 60\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 68 + 57\cdot 167 + 18\cdot 167^{2} + 63\cdot 167^{3} + 86\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$15$ |
| $21$ |
$2$ |
$(1,2)$ |
$-5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
