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:
| $r_{ 1 }$ |
$=$ |
$ 94 a + 118 + \left(120 a + 164\right)\cdot 167 + \left(106 a + 53\right)\cdot 167^{2} + \left(10 a + 81\right)\cdot 167^{3} + \left(101 a + 77\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 91 a + 49 + \left(3 a + 39\right)\cdot 167 + \left(103 a + 88\right)\cdot 167^{2} + \left(44 a + 26\right)\cdot 167^{3} + 102 a\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 79 + 4\cdot 167 + 141\cdot 167^{2} + 93\cdot 167^{3} + 147\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 40\cdot 167 + 154\cdot 167^{2} + 107\cdot 167^{3} + 80\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 109\cdot 167 + 2\cdot 167^{2} + 71\cdot 167^{3} + 136\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 73 a + 45 + \left(46 a + 24\right)\cdot 167 + \left(60 a + 40\right)\cdot 167^{2} + \left(156 a + 152\right)\cdot 167^{3} + 65 a\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 76 a + 140 + \left(163 a + 118\right)\cdot 167 + \left(63 a + 20\right)\cdot 167^{2} + \left(122 a + 135\right)\cdot 167^{3} + \left(64 a + 57\right)\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
