Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 18 + \left(16 a + 8\right)\cdot 83 + \left(63 a + 59\right)\cdot 83^{2} + \left(71 a + 78\right)\cdot 83^{3} + \left(50 a + 3\right)\cdot 83^{4} + \left(2 a + 11\right)\cdot 83^{5} + \left(78 a + 1\right)\cdot 83^{6} +O\left(83^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a + 19 + \left(4 a + 2\right)\cdot 83 + \left(68 a + 45\right)\cdot 83^{2} + \left(22 a + 14\right)\cdot 83^{3} + \left(76 a + 18\right)\cdot 83^{4} + \left(23 a + 44\right)\cdot 83^{5} + \left(56 a + 76\right)\cdot 83^{6} +O\left(83^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 75 a + 20 + \left(30 a + 73\right)\cdot 83 + \left(30 a + 6\right)\cdot 83^{2} + \left(48 a + 82\right)\cdot 83^{3} + \left(7 a + 64\right)\cdot 83^{4} + \left(53 a + 13\right)\cdot 83^{5} + \left(62 a + 71\right)\cdot 83^{6} +O\left(83^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 12 + \left(52 a + 29\right)\cdot 83 + \left(52 a + 6\right)\cdot 83^{2} + \left(34 a + 17\right)\cdot 83^{3} + \left(75 a + 24\right)\cdot 83^{4} + \left(29 a + 59\right)\cdot 83^{5} + \left(20 a + 80\right)\cdot 83^{6} +O\left(83^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 56 + \left(66 a + 69\right)\cdot 83 + \left(19 a + 22\right)\cdot 83^{2} + \left(11 a + 4\right)\cdot 83^{3} + \left(32 a + 66\right)\cdot 83^{4} + \left(80 a + 45\right)\cdot 83^{5} + \left(4 a + 76\right)\cdot 83^{6} +O\left(83^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 a + 42 + \left(78 a + 66\right)\cdot 83 + \left(14 a + 25\right)\cdot 83^{2} + \left(60 a + 52\right)\cdot 83^{3} + \left(6 a + 71\right)\cdot 83^{4} + \left(59 a + 74\right)\cdot 83^{5} + \left(26 a + 25\right)\cdot 83^{6} +O\left(83^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,6)$ |
| $(2,4,5)$ |
| $(1,2,6,4,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $1$ | $3$ | $(1,6,3)(2,4,5)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,3,6)(2,5,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,3,6)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,3,6)(2,4,5)$ | $-1$ |
| $3$ | $6$ | $(1,2,6,4,3,5)$ | $0$ |
| $3$ | $6$ | $(1,5,3,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
