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:
| $r_{ 1 }$ |
$=$ |
$ 59 a + 12 + \left(84 a + 33\right)\cdot 109 + \left(66 a + 60\right)\cdot 109^{2} + \left(75 a + 27\right)\cdot 109^{3} + \left(22 a + 74\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 17\cdot 109 + 6\cdot 109^{2} + 45\cdot 109^{3} + 13\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 69 a + 33 + \left(89 a + 96\right)\cdot 109 + \left(103 a + 28\right)\cdot 109^{2} + \left(6 a + 25\right)\cdot 109^{3} + \left(32 a + 55\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a + 102 + \left(19 a + 7\right)\cdot 109 + \left(5 a + 43\right)\cdot 109^{2} + \left(102 a + 37\right)\cdot 109^{3} + \left(76 a + 80\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 84 + 4\cdot 109 + 37\cdot 109^{2} + 46\cdot 109^{3} + 82\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 a + 71 + \left(24 a + 58\right)\cdot 109 + \left(42 a + 42\right)\cdot 109^{2} + \left(33 a + 36\right)\cdot 109^{3} + \left(86 a + 21\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)$ |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(2,6)$ | $-2$ |
| $9$ | $2$ | $(2,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,6)$ | $1$ |
| $4$ | $3$ | $(1,2,6)(3,4,5)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,5,6,3)$ | $0$ |
| $12$ | $6$ | $(2,6)(3,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
