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 + 99 + \left(77 a + 33\right)\cdot 109 + 108 a\cdot 109^{2} + \left(71 a + 26\right)\cdot 109^{3} + \left(84 a + 78\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 102 + \left(29 a + 107\right)\cdot 109 + \left(67 a + 17\right)\cdot 109^{2} + \left(18 a + 7\right)\cdot 109^{3} + \left(9 a + 77\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 89 a + 13 + \left(79 a + 8\right)\cdot 109 + \left(41 a + 56\right)\cdot 109^{2} + \left(90 a + 67\right)\cdot 109^{3} + \left(99 a + 67\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 a + 49 + \left(31 a + 52\right)\cdot 109 + 31\cdot 109^{2} + \left(37 a + 98\right)\cdot 109^{3} + \left(24 a + 90\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 85 + \left(62 a + 32\right)\cdot 109 + \left(80 a + 101\right)\cdot 109^{2} + \left(3 a + 47\right)\cdot 109^{3} + \left(83 a + 21\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 106 a + 88 + \left(46 a + 91\right)\cdot 109 + \left(28 a + 10\right)\cdot 109^{2} + \left(105 a + 80\right)\cdot 109^{3} + \left(25 a + 100\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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $15$ | $2$ | $(1,2)$ | $-3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
