Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{3} + 3 x + 81 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 a^{2} + 31 + \left(12 a + 21\right)\cdot 83 + \left(8 a^{2} + 45 a + 37\right)\cdot 83^{2} + \left(22 a + 82\right)\cdot 83^{3} + \left(13 a^{2} + 11 a + 72\right)\cdot 83^{4} + \left(70 a^{2} + 14 a + 56\right)\cdot 83^{5} + \left(55 a^{2} + 26 a + 18\right)\cdot 83^{6} + \left(20 a^{2} + 32 a + 40\right)\cdot 83^{7} + \left(19 a^{2} + 24 a + 59\right)\cdot 83^{8} + \left(29 a^{2} + 41 a + 38\right)\cdot 83^{9} + \left(23 a^{2} + a + 31\right)\cdot 83^{10} + \left(63 a^{2} + 40 a + 17\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 + 3\cdot 83 + 71\cdot 83^{2} + 32\cdot 83^{3} + 22\cdot 83^{4} + 71\cdot 83^{5} + 45\cdot 83^{6} + 61\cdot 83^{7} + 40\cdot 83^{8} + 21\cdot 83^{9} + 34\cdot 83^{10} + 67\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 + 60\cdot 83 + 36\cdot 83^{2} + 40\cdot 83^{3} + 30\cdot 83^{4} + 23\cdot 83^{5} + 23\cdot 83^{6} + 62\cdot 83^{7} + 83^{8} + 62\cdot 83^{9} + 52\cdot 83^{10} + 30\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 a^{2} + 19 a + 62 + \left(8 a^{2} + 64 a + 79\right)\cdot 83 + \left(42 a + 35\right)\cdot 83^{2} + \left(12 a^{2} + 19 a + 28\right)\cdot 83^{3} + \left(28 a^{2} + 35 a + 13\right)\cdot 83^{4} + \left(61 a^{2} + 28 a + 7\right)\cdot 83^{5} + \left(65 a^{2} + 48 a + 27\right)\cdot 83^{6} + \left(11 a^{2} + 67 a + 73\right)\cdot 83^{7} + \left(68 a^{2} + 7 a + 7\right)\cdot 83^{8} + \left(63 a^{2} + 42 a + 29\right)\cdot 83^{9} + \left(54 a^{2} + 55 a + 32\right)\cdot 83^{10} + \left(6 a^{2} + 48 a + 53\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a^{2} + 38 a + 49 + \left(78 a^{2} + 55 a + 11\right)\cdot 83 + \left(a^{2} + 43 a + 25\right)\cdot 83^{2} + \left(44 a^{2} + 6 a + 4\right)\cdot 83^{3} + \left(33 a^{2} + 12 a + 31\right)\cdot 83^{4} + \left(13 a^{2} + 38 a + 26\right)\cdot 83^{5} + \left(20 a^{2} + 76 a + 30\right)\cdot 83^{6} + \left(74 a^{2} + 43 a + 64\right)\cdot 83^{7} + \left(47 a^{2} + 33 a + 33\right)\cdot 83^{8} + \left(66 a^{2} + 61 a + 30\right)\cdot 83^{9} + \left(56 a^{2} + 57 a + 15\right)\cdot 83^{10} + \left(56 a^{2} + 7 a + 4\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a^{2} + 45 a + 35 + \left(4 a^{2} + 15 a + 29\right)\cdot 83 + \left(73 a^{2} + 77 a + 1\right)\cdot 83^{2} + \left(38 a^{2} + 53 a + 77\right)\cdot 83^{3} + \left(36 a^{2} + 59 a + 36\right)\cdot 83^{4} + \left(82 a^{2} + 30 a + 81\right)\cdot 83^{5} + \left(6 a^{2} + 63 a + 3\right)\cdot 83^{6} + \left(71 a^{2} + 6 a + 58\right)\cdot 83^{7} + \left(15 a^{2} + 25 a + 52\right)\cdot 83^{8} + \left(70 a^{2} + 63 a + 37\right)\cdot 83^{9} + \left(2 a^{2} + 23 a + 73\right)\cdot 83^{10} + \left(46 a^{2} + 35 a + 65\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 + 18\cdot 83 + 53\cdot 83^{2} + 81\cdot 83^{3} + 17\cdot 83^{4} + 4\cdot 83^{5} + 25\cdot 83^{6} + 62\cdot 83^{7} + 30\cdot 83^{8} + 22\cdot 83^{9} + 24\cdot 83^{10} + 26\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 78 a^{2} + 81 a + 15 + \left(3 a^{2} + 50 a + 70\right)\cdot 83 + \left(80 a^{2} + 20 a + 29\right)\cdot 83^{2} + \left(31 a^{2} + 34 a + 68\right)\cdot 83^{3} + \left(42 a^{2} + 73 a + 41\right)\cdot 83^{4} + \left(13 a^{2} + 14 a + 77\right)\cdot 83^{5} + \left(78 a^{2} + 61 a + 51\right)\cdot 83^{6} + \left(6 a^{2} + 20 a + 63\right)\cdot 83^{7} + \left(68 a^{2} + 56 a + 7\right)\cdot 83^{8} + \left(65 a^{2} + 51 a + 33\right)\cdot 83^{9} + \left(12 a^{2} + 81 a + 31\right)\cdot 83^{10} + \left(50 a + 40\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 28 a^{2} + 66 a + 81 + \left(70 a^{2} + 50 a + 36\right)\cdot 83 + \left(2 a^{2} + 19 a + 41\right)\cdot 83^{2} + \left(39 a^{2} + 29 a + 82\right)\cdot 83^{3} + \left(12 a^{2} + 57 a + 64\right)\cdot 83^{4} + \left(8 a^{2} + 39 a + 66\right)\cdot 83^{5} + \left(22 a^{2} + 56 a + 22\right)\cdot 83^{6} + \left(64 a^{2} + 77 a + 12\right)\cdot 83^{7} + \left(29 a^{2} + 18 a + 14\right)\cdot 83^{8} + \left(36 a^{2} + 72 a + 57\right)\cdot 83^{9} + \left(15 a^{2} + 28 a + 36\right)\cdot 83^{10} + \left(76 a^{2} + 66 a + 26\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,4)(3,8)(7,9)$ |
| $(1,6)(3,7)(4,8)$ |
| $(1,9,7)(2,6,4)(3,5,8)$ |
| $(1,6,5)(2,3,7)(4,8,9)$ |
| $(2,3,7)(4,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,5)(2,3)(4,9)$ | $0$ |
| $9$ | $2$ | $(2,4)(3,8)(7,9)$ | $-2$ |
| $9$ | $2$ | $(1,8)(3,7)(4,6)(5,9)$ | $0$ |
| $2$ | $3$ | $(1,6,5)(2,3,7)(4,8,9)$ | $-3$ |
| $6$ | $3$ | $(1,9,7)(2,6,4)(3,5,8)$ | $0$ |
| $6$ | $3$ | $(1,6,5)(4,9,8)$ | $0$ |
| $12$ | $3$ | $(1,9,3)(2,5,8)(4,7,6)$ | $0$ |
| $18$ | $6$ | $(1,9,3,5,4,2)(6,8,7)$ | $0$ |
| $18$ | $6$ | $(1,6,5)(2,8,7,4,3,9)$ | $1$ |
| $18$ | $6$ | $(1,9,6,8,5,4)(3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
