Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 44 a + 29 + \left(13 a + 17\right)\cdot 83 + \left(37 a + 32\right)\cdot 83^{2} + \left(29 a + 27\right)\cdot 83^{3} + \left(5 a + 43\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 12 + 74 a\cdot 83 + \left(75 a + 20\right)\cdot 83^{2} + \left(45 a + 16\right)\cdot 83^{3} + \left(23 a + 31\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 a + 13 + \left(8 a + 73\right)\cdot 83 + \left(7 a + 21\right)\cdot 83^{2} + \left(37 a + 69\right)\cdot 83^{3} + \left(59 a + 8\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 a + 73 + \left(50 a + 48\right)\cdot 83 + \left(31 a + 27\right)\cdot 83^{2} + \left(78 a + 76\right)\cdot 83^{3} + \left(27 a + 56\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 a + 73 + \left(69 a + 69\right)\cdot 83 + \left(45 a + 55\right)\cdot 83^{2} + \left(53 a + 19\right)\cdot 83^{3} + \left(77 a + 19\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 50 + \left(32 a + 39\right)\cdot 83 + \left(51 a + 8\right)\cdot 83^{2} + \left(4 a + 40\right)\cdot 83^{3} + \left(55 a + 6\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,2)(3,5,4)$ |
| $(1,5)(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,6,2)(3,5,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$6$ |
$(1,4,2,5,6,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,6,5,2,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
