Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{3} + 3 x + 86 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a^{2} + 21 a + 72 + \left(39 a^{2} + 83 a + 43\right)\cdot 89 + \left(16 a^{2} + 61 a + 81\right)\cdot 89^{2} + \left(74 a^{2} + 53 a + 13\right)\cdot 89^{3} + \left(52 a^{2} + 82 a + 17\right)\cdot 89^{4} + \left(35 a^{2} + 46 a + 59\right)\cdot 89^{5} + \left(26 a^{2} + 70 a + 87\right)\cdot 89^{6} + \left(52 a^{2} + 43 a + 18\right)\cdot 89^{7} + \left(54 a^{2} + 24 a + 28\right)\cdot 89^{8} + \left(14 a^{2} + 25 a + 64\right)\cdot 89^{9} + \left(28 a^{2} + 71 a + 15\right)\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 3\cdot 89 + 10\cdot 89^{2} + 58\cdot 89^{3} + 12\cdot 89^{4} + 22\cdot 89^{5} + 41\cdot 89^{6} + 16\cdot 89^{7} + 12\cdot 89^{8} + 25\cdot 89^{9} + 50\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 a^{2} + 87 a + 7 + \left(12 a^{2} + 68 a + 30\right)\cdot 89 + \left(19 a^{2} + 84 a + 65\right)\cdot 89^{2} + \left(32 a^{2} + 12 a + 82\right)\cdot 89^{3} + \left(25 a^{2} + 42 a + 41\right)\cdot 89^{4} + \left(45 a^{2} + 38 a + 2\right)\cdot 89^{5} + \left(66 a^{2} + 82 a + 52\right)\cdot 89^{6} + \left(59 a^{2} + 7 a + 39\right)\cdot 89^{7} + \left(19 a^{2} + 73 a + 12\right)\cdot 89^{8} + \left(88 a^{2} + 57 a + 71\right)\cdot 89^{9} + \left(6 a^{2} + 31 a + 67\right)\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a^{2} + 87 a + 62 + \left(28 a^{2} + 84 a + 61\right)\cdot 89 + \left(35 a^{2} + 54 a + 8\right)\cdot 89^{2} + \left(65 a^{2} + 14 a + 60\right)\cdot 89^{3} + \left(32 a^{2} + 48 a + 56\right)\cdot 89^{4} + \left(53 a^{2} + 27 a + 18\right)\cdot 89^{5} + \left(59 a^{2} + 77 a + 38\right)\cdot 89^{6} + \left(88 a^{2} + 40 a + 8\right)\cdot 89^{7} + \left(75 a^{2} + 46 a + 36\right)\cdot 89^{8} + \left(63 a^{2} + 8 a + 22\right)\cdot 89^{9} + \left(78 a^{2} + 12 a + 33\right)\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 83 a^{2} + 4 a + 61 + \left(47 a^{2} + 24 a + 11\right)\cdot 89 + \left(34 a^{2} + 38 a + 7\right)\cdot 89^{2} + \left(80 a^{2} + 61 a + 1\right)\cdot 89^{3} + \left(30 a^{2} + 87 a + 53\right)\cdot 89^{4} + \left(79 a^{2} + 22 a + 70\right)\cdot 89^{5} + \left(51 a^{2} + 18 a + 22\right)\cdot 89^{6} + \left(29 a^{2} + 40 a + 68\right)\cdot 89^{7} + \left(82 a^{2} + 58 a + 48\right)\cdot 89^{8} + \left(25 a^{2} + 22 a + 35\right)\cdot 89^{9} + \left(3 a^{2} + 45 a + 60\right)\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 79\cdot 89 + 69\cdot 89^{2} + 46\cdot 89^{3} + 5\cdot 89^{4} + 22\cdot 89^{5} + 67\cdot 89^{6} + 41\cdot 89^{7} + 25\cdot 89^{8} + 48\cdot 89^{9} + 81\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 79 + 6\cdot 89 + 49\cdot 89^{2} + 65\cdot 89^{3} + 6\cdot 89^{4} + 78\cdot 89^{5} + 29\cdot 89^{6} + 82\cdot 89^{7} + 18\cdot 89^{8} + 48\cdot 89^{9} + 6\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 13 a^{2} + 26 a + 58 + \left(51 a^{2} + 24 a + 67\right)\cdot 89 + \left(84 a^{2} + 61 a + 39\right)\cdot 89^{2} + \left(82 a^{2} + 55 a + 31\right)\cdot 89^{3} + \left(35 a^{2} + 66 a + 72\right)\cdot 89^{4} + \left(56 a^{2} + 85 a + 11\right)\cdot 89^{5} + \left(73 a^{2} + 63 a + 4\right)\cdot 89^{6} + \left(30 a^{2} + 51 a + 65\right)\cdot 89^{7} + \left(79 a^{2} + 32 a + 77\right)\cdot 89^{8} + \left(40 a^{2} + 12 a + 27\right)\cdot 89^{9} + \left(33 a^{2} + 72 a + 26\right)\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 56 a^{2} + 42 a + 55 + \left(87 a^{2} + 70 a + 51\right)\cdot 89 + \left(76 a^{2} + 54 a + 24\right)\cdot 89^{2} + \left(20 a^{2} + 68 a + 85\right)\cdot 89^{3} + 28 a\cdot 89^{4} + \left(86 a^{2} + 45 a + 71\right)\cdot 89^{5} + \left(77 a^{2} + 43 a + 12\right)\cdot 89^{6} + \left(5 a^{2} + 82 a + 15\right)\cdot 89^{7} + \left(44 a^{2} + 31 a + 7\right)\cdot 89^{8} + \left(33 a^{2} + 51 a + 13\right)\cdot 89^{9} + \left(27 a^{2} + 34 a + 14\right)\cdot 89^{10} +O\left(89^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,8,9)(2,7,6)$ |
| $(1,8,9)(3,5,4)$ |
| $(1,8)(2,6)(3,4)$ |
| $(1,3)(4,8)(5,9)$ |
| $(1,3,2)(4,6,8)(5,7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,8)(2,6)(3,4)$ | $0$ |
| $9$ | $2$ | $(1,3)(4,8)(5,9)$ | $-2$ |
| $9$ | $2$ | $(1,4)(2,6)(3,8)(5,9)$ | $0$ |
| $2$ | $3$ | $(1,8,9)(2,6,7)(3,4,5)$ | $-3$ |
| $6$ | $3$ | $(1,3,2)(4,6,8)(5,7,9)$ | $0$ |
| $6$ | $3$ | $(1,8,9)(2,7,6)$ | $0$ |
| $12$ | $3$ | $(1,3,7)(2,8,4)(5,6,9)$ | $0$ |
| $18$ | $6$ | $(1,4,2,8,3,6)(5,7,9)$ | $0$ |
| $18$ | $6$ | $(1,3,8,4,9,5)(2,7,6)$ | $1$ |
| $18$ | $6$ | $(1,4,8,3,9,5)(6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
