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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 106 + \left(13 a + 43\right)\cdot 107 + \left(15 a + 56\right)\cdot 107^{2} + \left(77 a + 47\right)\cdot 107^{3} + \left(64 a + 80\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 77 a + 1 + \left(46 a + 54\right)\cdot 107 + \left(60 a + 51\right)\cdot 107^{2} + \left(41 a + 52\right)\cdot 107^{3} + \left(6 a + 34\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 + 80\cdot 107 + 53\cdot 107^{2} + 39\cdot 107^{3} + 85\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 a + 95 + \left(60 a + 56\right)\cdot 107 + \left(46 a + 32\right)\cdot 107^{2} + \left(65 a + 51\right)\cdot 107^{3} + \left(100 a + 18\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 100 a + 27 + \left(93 a + 89\right)\cdot 107 + \left(91 a + 103\right)\cdot 107^{2} + \left(29 a + 19\right)\cdot 107^{3} + \left(42 a + 48\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 + 103\cdot 107 + 22\cdot 107^{2} + 3\cdot 107^{3} + 54\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)$ |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $-2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,3,5)$ | $1$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $0$ |
| $12$ | $6$ | $(2,4,6)(3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
