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:
| $r_{ 1 }$ |
$=$ |
$ 73 a + 53 + \left(61 a + 1\right)\cdot 83 + \left(18 a + 34\right)\cdot 83^{2} + \left(11 a + 4\right)\cdot 83^{3} + \left(46 a + 30\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 71 + 7\cdot 83 + 58\cdot 83^{2} + 81\cdot 83^{3} + 70\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 6\cdot 83 + 81\cdot 83^{2} + 10\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 43 + \left(21 a + 73\right)\cdot 83 + \left(64 a + 73\right)\cdot 83^{2} + \left(71 a + 79\right)\cdot 83^{3} + \left(36 a + 64\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 a + 81 + \left(23 a + 60\right)\cdot 83 + \left(13 a + 47\right)\cdot 83^{2} + \left(48 a + 23\right)\cdot 83^{3} + \left(36 a + 42\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 67 + \left(59 a + 15\right)\cdot 83 + \left(69 a + 37\right)\cdot 83^{2} + \left(34 a + 58\right)\cdot 83^{3} + \left(46 a + 30\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(1,2)$ | $2$ |
| $9$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $4$ | $3$ | $(3,5,6)$ | $1$ |
| $18$ | $4$ | $(1,5,2,3)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,3,2,5,4,6)$ | $0$ |
| $12$ | $6$ | $(1,2)(3,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
