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:
| $r_{ 1 }$ |
$=$ |
$ 107 + 10\cdot 109 + 91\cdot 109^{2} + 36\cdot 109^{3} + 11\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 98 a + 102 + \left(82 a + 9\right)\cdot 109 + \left(100 a + 85\right)\cdot 109^{2} + \left(67 a + 56\right)\cdot 109^{3} + \left(34 a + 71\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 91 + \left(26 a + 103\right)\cdot 109 + \left(8 a + 102\right)\cdot 109^{2} + \left(41 a + 23\right)\cdot 109^{3} + \left(74 a + 38\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 105 + 20\cdot 109 + 56\cdot 109^{2} + 36\cdot 109^{3} + 86\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 60 + \left(108 a + 101\right)\cdot 109 + \left(33 a + 32\right)\cdot 109^{2} + 103\cdot 109^{3} + \left(43 a + 92\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 88 a + 81 + 79\cdot 109 + \left(75 a + 67\right)\cdot 109^{2} + \left(108 a + 69\right)\cdot 109^{3} + \left(65 a + 26\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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(2,3)$ | $-2$ |
| $9$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $4$ | $3$ | $(2,3,4)$ | $1$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $-2$ |
| $18$ | $4$ | $(1,2,5,3)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,2,5,3,6,4)$ | $0$ |
| $12$ | $6$ | $(1,5,6)(2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
