Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{3} + 3 x + 81 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 a^{2} + 10 a + 31 + \left(10 a^{2} + 2 a + 49\right)\cdot 83 + \left(79 a^{2} + 40 a + 68\right)\cdot 83^{2} + \left(75 a^{2} + 9 a + 34\right)\cdot 83^{3} + \left(53 a^{2} + 45 a + 26\right)\cdot 83^{4} + \left(64 a^{2} + 50 a + 5\right)\cdot 83^{5} + \left(53 a^{2} + 61 a + 26\right)\cdot 83^{6} + \left(81 a^{2} + 81 a + 69\right)\cdot 83^{7} + \left(26 a^{2} + a + 66\right)\cdot 83^{8} + \left(57 a^{2} + 79 a + 82\right)\cdot 83^{9} +O\left(83^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 a^{2} + 31 a + 38 + \left(37 a^{2} + 69 a + 21\right)\cdot 83 + \left(49 a^{2} + 20 a + 9\right)\cdot 83^{2} + \left(79 a^{2} + 51 a + 42\right)\cdot 83^{3} + \left(52 a^{2} + 63 a + 24\right)\cdot 83^{4} + \left(2 a^{2} + 28 a + 47\right)\cdot 83^{5} + \left(39 a^{2} + 78 a + 79\right)\cdot 83^{6} + \left(74 a^{2} + 6 a + 54\right)\cdot 83^{7} + \left(53 a^{2} + 41 a + 37\right)\cdot 83^{8} + \left(21 a^{2} + 12 a + 11\right)\cdot 83^{9} +O\left(83^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 51\cdot 83 + 54\cdot 83^{2} + 45\cdot 83^{3} + 28\cdot 83^{4} + 49\cdot 83^{5} + 66\cdot 83^{6} + 41\cdot 83^{7} + 68\cdot 83^{8} + 51\cdot 83^{9} +O\left(83^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 73 a^{2} + 45 a + 66 + \left(65 a^{2} + 30 a + 1\right)\cdot 83 + \left(54 a^{2} + 71 a + 37\right)\cdot 83^{2} + \left(12 a^{2} + 64 a + 16\right)\cdot 83^{3} + \left(22 a^{2} + 2 a + 19\right)\cdot 83^{4} + \left(60 a^{2} + 17 a + 32\right)\cdot 83^{5} + \left(74 a^{2} + 60 a + 8\right)\cdot 83^{6} + \left(5 a^{2} + 74 a + 18\right)\cdot 83^{7} + \left(63 a^{2} + 29 a + 82\right)\cdot 83^{8} + \left(48 a^{2} + 54 a + 3\right)\cdot 83^{9} +O\left(83^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a^{2} + 42 a + \left(35 a^{2} + 11 a + 17\right)\cdot 83 + \left(37 a^{2} + 22 a + 68\right)\cdot 83^{2} + \left(10 a^{2} + 22 a + 69\right)\cdot 83^{3} + \left(59 a^{2} + 57 a + 36\right)\cdot 83^{4} + \left(15 a^{2} + 3 a + 73\right)\cdot 83^{5} + \left(73 a^{2} + 26 a + 64\right)\cdot 83^{6} + \left(9 a^{2} + 77 a + 8\right)\cdot 83^{7} + \left(2 a^{2} + 39 a + 17\right)\cdot 83^{8} + \left(4 a^{2} + 74 a + 59\right)\cdot 83^{9} +O\left(83^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a^{2} + 26 a + 35 + \left(20 a^{2} + 14 a + 76\right)\cdot 83 + \left(7 a^{2} + 73 a + 24\right)\cdot 83^{2} + \left(48 a^{2} + 53 a + 4\right)\cdot 83^{3} + \left(11 a^{2} + 63 a + 81\right)\cdot 83^{4} + \left(9 a^{2} + 44 a + 12\right)\cdot 83^{5} + \left(21 a^{2} + 68 a + 67\right)\cdot 83^{6} + \left(18 a^{2} + 58 a + 42\right)\cdot 83^{7} + \left(26 a^{2} + 36 a + 8\right)\cdot 83^{8} + \left(80 a^{2} + 77 a + 67\right)\cdot 83^{9} +O\left(83^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 77 a^{2} + 12 a + 74 + \left(79 a^{2} + 38 a + 29\right)\cdot 83 + \left(20 a^{2} + 21 a + 52\right)\cdot 83^{2} + \left(22 a^{2} + 47 a + 35\right)\cdot 83^{3} + \left(49 a^{2} + 16 a + 73\right)\cdot 83^{4} + \left(13 a^{2} + 21 a + 21\right)\cdot 83^{5} + \left(70 a^{2} + 37 a + 82\right)\cdot 83^{6} + \left(58 a^{2} + 32 a + 40\right)\cdot 83^{7} + \left(76 a^{2} + 16 a + 26\right)\cdot 83^{8} + \left(36 a^{2} + 34 a + 63\right)\cdot 83^{9} +O\left(83^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 12 + 19\cdot 83 + 73\cdot 83^{2} + 80\cdot 83^{3} + 37\cdot 83^{4} + 7\cdot 83^{5} + 2\cdot 83^{6} + 27\cdot 83^{7} + 13\cdot 83^{8} + 37\cdot 83^{9} +O\left(83^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 52 + 65\cdot 83 + 26\cdot 83^{2} + 2\cdot 83^{3} + 4\cdot 83^{4} + 82\cdot 83^{5} + 17\cdot 83^{6} + 28\cdot 83^{7} + 11\cdot 83^{8} + 38\cdot 83^{9} +O\left(83^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,7,8)(2,6,3)(4,9,5)$ |
| $(1,2,5)(4,6,7)$ |
| $(1,2)(3,7)(4,9)(6,8)$ |
| $(3,9,8)(4,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,2)(3,7)(4,9)(6,8)$ | $0$ |
| $2$ | $3$ | $(1,2,5)(3,9,8)(4,7,6)$ | $-3$ |
| $3$ | $3$ | $(1,2,5)(4,6,7)$ | $0$ |
| $3$ | $3$ | $(1,5,2)(4,7,6)$ | $0$ |
| $6$ | $3$ | $(1,7,8)(2,6,3)(4,9,5)$ | $0$ |
| $6$ | $3$ | $(1,4,9)(2,7,8)(3,5,6)$ | $0$ |
| $6$ | $3$ | $(1,9,4)(2,8,7)(3,6,5)$ | $0$ |
| $9$ | $6$ | $(1,5)(3,4,9,6,8,7)$ | $0$ |
| $9$ | $6$ | $(1,5)(3,7,8,6,9,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
