Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{2} + 103 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 a + 99 + \left(84 a + 30\right)\cdot 107 + \left(67 a + 79\right)\cdot 107^{2} + \left(23 a + 76\right)\cdot 107^{3} + \left(13 a + 62\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a + 77 + \left(53 a + 44\right)\cdot 107 + \left(14 a + 26\right)\cdot 107^{2} + \left(69 a + 72\right)\cdot 107^{3} + \left(60 a + 57\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 75 a + 13 + \left(22 a + 15\right)\cdot 107 + \left(39 a + 52\right)\cdot 107^{2} + \left(83 a + 103\right)\cdot 107^{3} + \left(93 a + 91\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 75 + 41\cdot 107 + 49\cdot 107^{2} + 21\cdot 107^{3} + 32\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 84 a + 62 + \left(53 a + 20\right)\cdot 107 + \left(92 a + 31\right)\cdot 107^{2} + \left(37 a + 13\right)\cdot 107^{3} + \left(46 a + 17\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 102 + 60\cdot 107 + 82\cdot 107^{2} + 33\cdot 107^{3} + 59\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $4$ | $3$ | $(1,3,6)$ | $-2$ |
| $18$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $1$ |
| $12$ | $6$ | $(1,3,6)(2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
