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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 56\cdot 83 + 63\cdot 83^{2} + 45\cdot 83^{3} + 6\cdot 83^{4} + 42\cdot 83^{5} + 79\cdot 83^{6} + 24\cdot 83^{7} + 16\cdot 83^{8} + 35\cdot 83^{9} + 46\cdot 83^{10} + 8\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a^{2} + 35 a + 66 + \left(11 a^{2} + 26 a + 77\right)\cdot 83 + \left(69 a^{2} + 12 a + 82\right)\cdot 83^{2} + \left(3 a^{2} + 58 a + 62\right)\cdot 83^{3} + \left(58 a^{2} + 62 a + 60\right)\cdot 83^{4} + \left(48 a^{2} + 69\right)\cdot 83^{5} + \left(18 a^{2} + 11 a + 64\right)\cdot 83^{6} + \left(11 a^{2} + 53 a + 77\right)\cdot 83^{7} + \left(80 a^{2} + 33 a + 21\right)\cdot 83^{8} + \left(44 a^{2} + 46 a + 62\right)\cdot 83^{9} + \left(79 a^{2} + 67 a + 20\right)\cdot 83^{10} + \left(40 a^{2} + 49 a + 54\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 a^{2} + 3 a + 67 + \left(24 a^{2} + 32 a + 21\right)\cdot 83 + \left(4 a^{2} + 6 a + 36\right)\cdot 83^{2} + \left(72 a^{2} + 65 a + 33\right)\cdot 83^{3} + \left(62 a^{2} + 73 a + 70\right)\cdot 83^{4} + \left(54 a^{2} + 76 a + 81\right)\cdot 83^{5} + \left(43 a^{2} + 29 a + 31\right)\cdot 83^{6} + \left(41 a^{2} + 39 a + 55\right)\cdot 83^{7} + \left(33 a^{2} + 31 a + 11\right)\cdot 83^{8} + \left(81 a^{2} + 13 a + 52\right)\cdot 83^{9} + \left(26 a^{2} + 7 a + 81\right)\cdot 83^{10} + \left(8 a^{2} + 48 a + 71\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 63\cdot 83 + 54\cdot 83^{2} + 58\cdot 83^{3} + 36\cdot 83^{4} + 74\cdot 83^{5} + 78\cdot 83^{6} + 37\cdot 83^{7} + 14\cdot 83^{8} + 38\cdot 83^{9} + 45\cdot 83^{10} + 41\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 65 + 46\cdot 83 + 47\cdot 83^{2} + 61\cdot 83^{3} + 39\cdot 83^{4} + 49\cdot 83^{5} + 7\cdot 83^{6} + 20\cdot 83^{7} + 52\cdot 83^{8} + 9\cdot 83^{9} + 74\cdot 83^{10} + 32\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a^{2} + 68 a + 32 + \left(58 a^{2} + 19 a + 5\right)\cdot 83 + \left(81 a^{2} + 16 a + 25\right)\cdot 83^{2} + \left(82 a^{2} + 20 a + 55\right)\cdot 83^{3} + \left(42 a^{2} + 25 a + 30\right)\cdot 83^{4} + \left(48 a^{2} + 52 a + 69\right)\cdot 83^{5} + \left(28 a^{2} + 59 a + 1\right)\cdot 83^{6} + \left(66 a^{2} + 81 a + 22\right)\cdot 83^{7} + \left(59 a^{2} + 11 a + 64\right)\cdot 83^{8} + \left(35 a^{2} + 62 a + 43\right)\cdot 83^{9} + \left(44 a^{2} + 69 a + 33\right)\cdot 83^{10} + \left(28 a^{2} + 35 a + 29\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 16 a^{2} + 3 a + 60 + \left(53 a^{2} + 14 a + 78\right)\cdot 83 + \left(45 a^{2} + 72 a + 35\right)\cdot 83^{2} + \left(52 a^{2} + 2 a + 77\right)\cdot 83^{3} + \left(15 a^{2} + 82 a + 58\right)\cdot 83^{4} + \left(27 a^{2} + 37 a + 26\right)\cdot 83^{5} + \left(4 a^{2} + 36 a + 36\right)\cdot 83^{6} + \left(32 a^{2} + 49 a + 36\right)\cdot 83^{7} + \left(63 a^{2} + 74 a + 71\right)\cdot 83^{8} + \left(22 a^{2} + 53 a + 17\right)\cdot 83^{9} + \left(31 a^{2} + 72 a + 7\right)\cdot 83^{10} + \left(43 a^{2} + 47 a + 59\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 6 a^{2} + 77 a + 40 + \left(5 a^{2} + 36 a + 65\right)\cdot 83 + \left(33 a^{2} + 4 a + 10\right)\cdot 83^{2} + \left(41 a^{2} + 15 a + 55\right)\cdot 83^{3} + \left(4 a^{2} + 10 a + 36\right)\cdot 83^{4} + \left(a^{2} + 51 a + 57\right)\cdot 83^{5} + \left(35 a^{2} + 16 a + 14\right)\cdot 83^{6} + \left(9 a^{2} + 77 a + 74\right)\cdot 83^{7} + \left(69 a^{2} + 59 a + 82\right)\cdot 83^{8} + \left(61 a^{2} + 15 a + 12\right)\cdot 83^{9} + \left(24 a^{2} + 3 a + 77\right)\cdot 83^{10} + \left(31 a^{2} + 70 a + 34\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 62 a^{2} + 63 a + 69 + \left(13 a^{2} + 36 a + 82\right)\cdot 83 + \left(15 a^{2} + 54 a + 57\right)\cdot 83^{2} + \left(79 a^{2} + 4 a + 47\right)\cdot 83^{3} + \left(64 a^{2} + 78 a + 74\right)\cdot 83^{4} + \left(68 a^{2} + 29 a + 26\right)\cdot 83^{5} + \left(35 a^{2} + 12 a + 16\right)\cdot 83^{6} + \left(5 a^{2} + 31 a + 66\right)\cdot 83^{7} + \left(26 a^{2} + 37 a + 79\right)\cdot 83^{8} + \left(2 a^{2} + 57 a + 59\right)\cdot 83^{9} + \left(42 a^{2} + 28 a + 28\right)\cdot 83^{10} + \left(13 a^{2} + 80 a + 82\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,6)(2,4,7)(5,8,9)$ |
| $(1,5,4)(3,7,8)$ |
| $(1,7)(2,6)(3,4)(5,8)$ |
| $(1,4,5)(2,9,6)(3,7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,7)(2,6)(3,4)(5,8)$ |
$0$ |
| $2$ |
$3$ |
$(1,4,5)(2,9,6)(3,7,8)$ |
$-3$ |
| $3$ |
$3$ |
$(1,5,4)(2,9,6)$ |
$0$ |
| $3$ |
$3$ |
$(1,4,5)(2,6,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,6)(2,4,7)(5,8,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,7,2)(3,6,5)(4,8,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,2,7)(3,5,6)(4,9,8)$ |
$0$ |
| $9$ |
$6$ |
$(1,2,5,9,4,6)(3,8)$ |
$0$ |
| $9$ |
$6$ |
$(1,6,4,9,5,2)(3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
