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:
| $r_{ 1 }$ |
$=$ |
$ 74 a + 19 + \left(48 a + 3\right)\cdot 83 + \left(47 a + 76\right)\cdot 83^{2} + \left(53 a + 3\right)\cdot 83^{3} + \left(67 a + 46\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 59 + \left(74 a + 21\right)\cdot 83 + 74\cdot 83^{2} + \left(4 a + 79\right)\cdot 83^{3} + \left(45 a + 81\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 + 51\cdot 83 + 7\cdot 83^{2} + 3\cdot 83^{3} + 44\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 10 + \left(34 a + 61\right)\cdot 83 + \left(35 a + 74\right)\cdot 83^{2} + \left(29 a + 9\right)\cdot 83^{3} + \left(15 a + 60\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 54 + 18\cdot 83 + 15\cdot 83^{2} + 69\cdot 83^{3} + 59\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 a + 62 + \left(8 a + 9\right)\cdot 83 + \left(82 a + 1\right)\cdot 83^{2} + 78 a\cdot 83^{3} + \left(37 a + 40\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(3,6)$ | $0$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
| $4$ | $3$ | $(1,4,5)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,6,5,2)$ | $-1$ |
| $12$ | $6$ | $(1,4,5)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
