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:
| $r_{ 1 }$ |
$=$ |
$ 84 a + 78 + \left(46 a + 22\right)\cdot 109 + \left(33 a + 89\right)\cdot 109^{2} + \left(5 a + 82\right)\cdot 109^{3} + \left(7 a + 105\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 53 + \left(62 a + 94\right)\cdot 109 + \left(75 a + 75\right)\cdot 109^{2} + \left(103 a + 54\right)\cdot 109^{3} + \left(101 a + 107\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 a + 45 + \left(108 a + 78\right)\cdot 109 + \left(2 a + 38\right)\cdot 109^{2} + \left(77 a + 95\right)\cdot 109^{3} + \left(47 a + 69\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 87 + 100\cdot 109 + 52\cdot 109^{2} + 80\cdot 109^{3} + 4\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 + 24\cdot 109 + 28\cdot 109^{2} + 62\cdot 109^{3} + 107\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a + 8 + 6\cdot 109 + \left(106 a + 42\right)\cdot 109^{2} + \left(31 a + 60\right)\cdot 109^{3} + \left(61 a + 40\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(1,2)$ | $2$ |
| $9$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $4$ | $3$ | $(3,5,6)$ | $1$ |
| $18$ | $4$ | $(1,5,2,3)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,3,2,5,4,6)$ | $0$ |
| $12$ | $6$ | $(1,2)(3,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
