Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{3} + 3 x + 86 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 78\cdot 89 + 7\cdot 89^{2} + 22\cdot 89^{3} + 86\cdot 89^{4} + 58\cdot 89^{5} + 50\cdot 89^{6} + 26\cdot 89^{7} + 55\cdot 89^{8} + 38\cdot 89^{9} + 31\cdot 89^{10} + 3\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a^{2} + 75 a + 12 + \left(12 a^{2} + 42 a + 33\right)\cdot 89 + \left(73 a^{2} + 68 a + 81\right)\cdot 89^{2} + \left(23 a^{2} + 47 a + 79\right)\cdot 89^{3} + \left(a^{2} + 80 a + 55\right)\cdot 89^{4} + \left(27 a^{2} + 35 a + 24\right)\cdot 89^{5} + \left(82 a^{2} + 62 a + 32\right)\cdot 89^{6} + \left(59 a^{2} + 27 a + 86\right)\cdot 89^{7} + \left(75 a^{2} + 38 a + 8\right)\cdot 89^{8} + \left(26 a^{2} + 81 a + 8\right)\cdot 89^{9} + \left(36 a^{2} + 40 a + 49\right)\cdot 89^{10} + \left(20 a^{2} + 70 a + 16\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 a^{2} + 6 a + 62 + \left(30 a^{2} + 69 a + 69\right)\cdot 89 + \left(34 a^{2} + 39 a + 3\right)\cdot 89^{2} + \left(43 a^{2} + 25 a + 30\right)\cdot 89^{3} + \left(28 a^{2} + 69 a + 21\right)\cdot 89^{4} + \left(75 a^{2} + 60 a + 32\right)\cdot 89^{5} + \left(8 a^{2} + 59 a + 63\right)\cdot 89^{6} + \left(37 a^{2} + 47 a + 40\right)\cdot 89^{7} + \left(22 a^{2} + 37 a + 80\right)\cdot 89^{8} + \left(69 a^{2} + 9 a + 3\right)\cdot 89^{9} + \left(28 a^{2} + 9 a + 34\right)\cdot 89^{10} + \left(68 a^{2} + 25 a + 23\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 54 a^{2} + 53 a + 42 + \left(83 a^{2} + 22 a + 28\right)\cdot 89 + \left(81 a^{2} + 47 a + 5\right)\cdot 89^{2} + \left(60 a^{2} + 2 a + 2\right)\cdot 89^{3} + \left(6 a^{2} + 58 a + 32\right)\cdot 89^{4} + \left(34 a^{2} + 6 a + 61\right)\cdot 89^{5} + \left(74 a^{2} + 79 a + 64\right)\cdot 89^{6} + \left(41 a^{2} + 76 a + 55\right)\cdot 89^{7} + \left(67 a^{2} + 22 a + 57\right)\cdot 89^{8} + \left(3 a^{2} + 40 a + 30\right)\cdot 89^{9} + \left(38 a^{2} + 13 a + 55\right)\cdot 89^{10} + \left(27 a^{2} + 10 a + 18\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 12\cdot 89 + 68\cdot 89^{2} + 48\cdot 89^{3} + 71\cdot 89^{4} + 48\cdot 89^{5} + 12\cdot 89^{6} + 81\cdot 89^{7} + 24\cdot 89^{8} + 5\cdot 89^{9} + 35\cdot 89^{10} + 13\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a^{2} + 18 a + 33 + \left(39 a^{2} + 18 a + 28\right)\cdot 89 + \left(17 a^{2} + 16 a + 54\right)\cdot 89^{2} + \left(83 a^{2} + 27 a + 46\right)\cdot 89^{3} + \left(22 a^{2} + 58 a + 64\right)\cdot 89^{4} + \left(48 a^{2} + 15 a\right)\cdot 89^{5} + \left(32 a^{2} + 52 a + 70\right)\cdot 89^{6} + \left(6 a^{2} + 17 a + 73\right)\cdot 89^{7} + \left(16 a^{2} + 39 a + 43\right)\cdot 89^{8} + \left(30 a^{2} + 60 a + 83\right)\cdot 89^{9} + \left(50 a^{2} + 39 a + 79\right)\cdot 89^{10} + \left(54 a^{2} + 73 a + 72\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 30 a^{2} + 18 a + 83 + \left(55 a^{2} + 48 a + 60\right)\cdot 89 + \left(78 a^{2} + 25 a + 87\right)\cdot 89^{2} + \left(33 a^{2} + 59 a + 36\right)\cdot 89^{3} + \left(59 a^{2} + 61 a + 48\right)\cdot 89^{4} + \left(6 a^{2} + 66 a + 6\right)\cdot 89^{5} + \left(71 a^{2} + 46 a + 58\right)\cdot 89^{6} + \left(40 a^{2} + 83 a + 53\right)\cdot 89^{7} + \left(5 a^{2} + 26 a + 22\right)\cdot 89^{8} + \left(55 a^{2} + 77 a + 44\right)\cdot 89^{9} + \left(35 a + 69\right)\cdot 89^{10} + \left(7 a^{2} + 5 a + 66\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 15 + 32\cdot 89 + 60\cdot 89^{2} + 13\cdot 89^{3} + 71\cdot 89^{4} + 52\cdot 89^{6} + 76\cdot 89^{7} + 44\cdot 89^{8} + 23\cdot 89^{9} + 66\cdot 89^{10} + 75\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 46 a^{2} + 8 a + 86 + \left(46 a^{2} + 66 a + 12\right)\cdot 89 + \left(70 a^{2} + 69 a + 76\right)\cdot 89^{2} + \left(21 a^{2} + 15 a + 75\right)\cdot 89^{3} + \left(59 a^{2} + 28 a + 82\right)\cdot 89^{4} + \left(75 a^{2} + 81 a + 32\right)\cdot 89^{5} + \left(86 a^{2} + 55 a + 41\right)\cdot 89^{6} + \left(80 a^{2} + 13 a + 39\right)\cdot 89^{7} + \left(79 a^{2} + 13 a + 17\right)\cdot 89^{8} + \left(81 a^{2} + 87 a + 29\right)\cdot 89^{9} + \left(23 a^{2} + 38 a + 24\right)\cdot 89^{10} + \left(82 a + 65\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,8,5)(2,9,3)(4,7,6)$ |
| $(2,6)(3,4)(5,8)(7,9)$ |
| $(2,3,9)(4,7,6)$ |
| $(1,4,2)(3,5,6)(7,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,9)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,8,5)(2,9,3)(4,7,6)$ | $-3$ |
| $3$ | $3$ | $(1,8,5)(2,3,9)$ | $0$ |
| $3$ | $3$ | $(1,5,8)(2,9,3)$ | $0$ |
| $6$ | $3$ | $(1,4,2)(3,5,6)(7,9,8)$ | $0$ |
| $6$ | $3$ | $(1,2,4)(3,6,5)(7,8,9)$ | $0$ |
| $6$ | $3$ | $(1,6,2)(3,5,7)(4,9,8)$ | $0$ |
| $9$ | $6$ | $(1,3,8,9,5,2)(4,6)$ | $0$ |
| $9$ | $6$ | $(1,2,5,9,8,3)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
