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:
| $r_{ 1 }$ |
$=$ |
$ 62 a^{2} + 28 a + 1 + \left(66 a^{2} + 42 a + 31\right)\cdot 89 + \left(80 a^{2} + 43 a + 81\right)\cdot 89^{2} + \left(65 a^{2} + 12 a + 32\right)\cdot 89^{3} + \left(17 a^{2} + 24 a + 15\right)\cdot 89^{4} + \left(60 a^{2} + 26\right)\cdot 89^{5} + \left(70 a^{2} + 9 a + 12\right)\cdot 89^{6} + \left(28 a^{2} + 54 a + 77\right)\cdot 89^{7} + \left(21 a^{2} + 69 a + 10\right)\cdot 89^{8} + \left(11 a^{2} + 82 a + 27\right)\cdot 89^{9} + \left(35 a^{2} + 6 a + 39\right)\cdot 89^{10} + \left(57 a^{2} + 85 a + 24\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 77\cdot 89 + 69\cdot 89^{2} + 7\cdot 89^{3} + 87\cdot 89^{4} + 82\cdot 89^{5} + 81\cdot 89^{6} + 76\cdot 89^{7} + 9\cdot 89^{8} + 29\cdot 89^{9} + 67\cdot 89^{10} + 71\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 60 a^{2} + 54 a + 86 + \left(34 a^{2} + 55 a + 55\right)\cdot 89 + \left(2 a^{2} + 39 a + 13\right)\cdot 89^{2} + \left(72 a^{2} + 88 a + 45\right)\cdot 89^{3} + \left(24 a^{2} + 13 a + 29\right)\cdot 89^{4} + \left(62 a^{2} + 12 a + 30\right)\cdot 89^{5} + \left(39 a^{2} + 43 a + 39\right)\cdot 89^{6} + \left(80 a^{2} + 80 a + 2\right)\cdot 89^{7} + \left(4 a^{2} + 44 a + 67\right)\cdot 89^{8} + \left(84 a^{2} + 17 a + 83\right)\cdot 89^{9} + \left(31 a^{2} + 84 a + 32\right)\cdot 89^{10} + \left(18 a^{2} + 15 a + 35\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a^{2} + 83 a + 48 + \left(28 a^{2} + 79 a + 37\right)\cdot 89 + \left(68 a^{2} + 53 a + 86\right)\cdot 89^{2} + \left(26 a^{2} + 47 a + 61\right)\cdot 89^{3} + \left(54 a^{2} + 68 a + 42\right)\cdot 89^{4} + \left(29 a^{2} + 57 a + 63\right)\cdot 89^{5} + \left(10 a^{2} + 21 a + 3\right)\cdot 89^{6} + \left(64 a^{2} + 77 a + 63\right)\cdot 89^{7} + \left(9 a^{2} + 7 a + 65\right)\cdot 89^{8} + \left(18 a^{2} + 18 a + 4\right)\cdot 89^{9} + \left(14 a^{2} + 55 a + 6\right)\cdot 89^{10} + \left(19 a^{2} + 80 a + 73\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 + 55\cdot 89 + 45\cdot 89^{2} + 39\cdot 89^{3} + 67\cdot 89^{4} + 23\cdot 89^{5} + 64\cdot 89^{6} + 79\cdot 89^{7} + 64\cdot 89^{8} + 31\cdot 89^{9} + 39\cdot 89^{10} + 40\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 a^{2} + 79 a + 13 + \left(70 a^{2} + 38 a + 33\right)\cdot 89 + \left(52 a^{2} + 13 a + 55\right)\cdot 89^{2} + \left(43 a^{2} + 57 a + 6\right)\cdot 89^{3} + \left(69 a^{2} + 52 a + 73\right)\cdot 89^{4} + \left(41 a^{2} + 36 a + 87\right)\cdot 89^{5} + \left(9 a^{2} + 23 a + 1\right)\cdot 89^{6} + \left(41 a^{2} + 31 a + 17\right)\cdot 89^{7} + \left(13 a^{2} + a + 73\right)\cdot 89^{8} + \left(25 a^{2} + 77 a + 18\right)\cdot 89^{9} + \left(84 a^{2} + 60 a + 57\right)\cdot 89^{10} + \left(32 a^{2} + 53 a + 11\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 + 53\cdot 89 + 8\cdot 89^{2} + 46\cdot 89^{3} + 14\cdot 89^{4} + 74\cdot 89^{5} + 24\cdot 89^{6} + 69\cdot 89^{7} + 59\cdot 89^{8} + 19\cdot 89^{9} + 53\cdot 89^{10} + 54\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 16 a + 75 + \left(79 a^{2} + 59 a + 49\right)\cdot 89 + \left(56 a^{2} + 21 a + 63\right)\cdot 89^{2} + \left(18 a^{2} + 73 a + 45\right)\cdot 89^{3} + \left(54 a^{2} + 56 a + 42\right)\cdot 89^{4} + \left(17 a^{2} + 83 a + 39\right)\cdot 89^{5} + \left(69 a^{2} + 43 a + 32\right)\cdot 89^{6} + \left(72 a^{2} + 69 a + 80\right)\cdot 89^{7} + \left(65 a^{2} + 79 a + 88\right)\cdot 89^{8} + \left(45 a^{2} + 82 a + 59\right)\cdot 89^{9} + \left(79 a^{2} + 61 a + 47\right)\cdot 89^{10} + \left(36 a^{2} + 43 a + 19\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 56 a^{2} + 7 a + 78 + \left(76 a^{2} + 80 a + 50\right)\cdot 89 + \left(5 a^{2} + 5 a + 20\right)\cdot 89^{2} + \left(40 a^{2} + 77 a + 70\right)\cdot 89^{3} + \left(46 a^{2} + 50 a + 72\right)\cdot 89^{4} + \left(55 a^{2} + 76 a + 16\right)\cdot 89^{5} + \left(67 a^{2} + 36 a + 6\right)\cdot 89^{6} + \left(68 a^{2} + 43 a + 68\right)\cdot 89^{7} + \left(62 a^{2} + 63 a + 4\right)\cdot 89^{8} + \left(82 a^{2} + 77 a + 81\right)\cdot 89^{9} + \left(21 a^{2} + 86 a + 12\right)\cdot 89^{10} + \left(13 a^{2} + 76 a + 25\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,9,3)(2,5,7)$ |
| $(1,4,2)(3,8,5)(6,7,9)$ |
| $(1,3,9)(4,6,8)$ |
| $(1,8)(2,5)(3,4)(6,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,8)(2,5)(3,4)(6,9)$ | $0$ |
| $2$ | $3$ | $(1,9,3)(2,7,5)(4,6,8)$ | $-3$ |
| $3$ | $3$ | $(1,9,3)(2,5,7)$ | $0$ |
| $3$ | $3$ | $(1,3,9)(2,7,5)$ | $0$ |
| $6$ | $3$ | $(1,4,2)(3,8,5)(6,7,9)$ | $0$ |
| $6$ | $3$ | $(1,4,5)(2,9,6)(3,8,7)$ | $0$ |
| $6$ | $3$ | $(1,5,4)(2,6,9)(3,7,8)$ | $0$ |
| $9$ | $6$ | $(1,8,9,6,3,4)(2,7)$ | $0$ |
| $9$ | $6$ | $(1,4,3,6,9,8)(2,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
