Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $ x^{2} + 108 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 a + 105 + \left(43 a + 22\right)\cdot 109 + \left(95 a + 39\right)\cdot 109^{2} + \left(6 a + 4\right)\cdot 109^{3} + \left(25 a + 79\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 a + 105 + \left(99 a + 45\right)\cdot 109 + \left(69 a + 61\right)\cdot 109^{2} + \left(106 a + 65\right)\cdot 109^{3} + \left(69 a + 88\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 72\cdot 109 + 32\cdot 109^{2} + 5\cdot 109^{3} + 108\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 a + 37 + \left(9 a + 104\right)\cdot 109 + \left(39 a + 31\right)\cdot 109^{2} + \left(2 a + 102\right)\cdot 109^{3} + \left(39 a + 51\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 64\cdot 109 + 70\cdot 109^{2} + 15\cdot 109^{3} + 11\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 a + 45 + \left(65 a + 17\right)\cdot 109 + \left(13 a + 91\right)\cdot 109^{2} + \left(102 a + 24\right)\cdot 109^{3} + \left(83 a + 97\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(3,4)$ | $0$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
| $4$ | $3$ | $(1,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,5,4,6,2)$ | $-1$ |
| $12$ | $6$ | $(1,5,6)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
