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:
| $r_{ 1 }$ |
$=$ |
$ 97 a + 59 + \left(40 a + 17\right)\cdot 109 + \left(86 a + 40\right)\cdot 109^{2} + \left(2 a + 24\right)\cdot 109^{3} + \left(15 a + 40\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 80 + \left(a + 76\right)\cdot 109 + \left(92 a + 36\right)\cdot 109^{2} + \left(3 a + 18\right)\cdot 109^{3} + \left(82 a + 75\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 47 + \left(68 a + 70\right)\cdot 109 + \left(22 a + 85\right)\cdot 109^{2} + \left(106 a + 49\right)\cdot 109^{3} + \left(93 a + 52\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 93 a + 37 + \left(33 a + 74\right)\cdot 109 + \left(3 a + 33\right)\cdot 109^{2} + \left(82 a + 58\right)\cdot 109^{3} + \left(51 a + 72\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 104 a + 85 + \left(107 a + 72\right)\cdot 109 + \left(16 a + 18\right)\cdot 109^{2} + \left(105 a + 39\right)\cdot 109^{3} + \left(26 a + 44\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 21 + \left(75 a + 15\right)\cdot 109 + \left(105 a + 3\right)\cdot 109^{2} + \left(26 a + 28\right)\cdot 109^{3} + \left(57 a + 42\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
