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:
| $r_{ 1 }$ |
$=$ |
$ 58 a^{2} + 76 a + 74 + \left(77 a^{2} + 11 a + 76\right)\cdot 89 + \left(43 a^{2} + 17 a + 73\right)\cdot 89^{2} + \left(29 a^{2} + 57 a + 63\right)\cdot 89^{3} + \left(15 a^{2} + 81 a + 17\right)\cdot 89^{4} + \left(10 a^{2} + 43 a + 86\right)\cdot 89^{5} + \left(78 a^{2} + 43 a + 31\right)\cdot 89^{6} + \left(31 a^{2} + 66 a + 76\right)\cdot 89^{7} + \left(33 a^{2} + 44 a + 33\right)\cdot 89^{8} + \left(13 a^{2} + 71 a + 49\right)\cdot 89^{9} + \left(5 a^{2} + 56 a + 53\right)\cdot 89^{10} + \left(28 a^{2} + 61 a + 28\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 72 + 79\cdot 89 + 41\cdot 89^{2} + 40\cdot 89^{3} + 4\cdot 89^{4} + 68\cdot 89^{5} + 80\cdot 89^{6} + 85\cdot 89^{7} + 48\cdot 89^{8} + 72\cdot 89^{9} + 32\cdot 89^{10} +O\left(89^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 60 a^{2} + 49 a + 22 + \left(79 a^{2} + 4 a + 82\right)\cdot 89 + \left(10 a^{2} + 66 a + 64\right)\cdot 89^{2} + \left(38 a^{2} + 88 a + 61\right)\cdot 89^{3} + \left(54 a^{2} + 3 a + 68\right)\cdot 89^{4} + \left(56 a^{2} + 12 a + 12\right)\cdot 89^{5} + \left(19 a^{2} + 26 a + 24\right)\cdot 89^{6} + \left(63 a^{2} + 52 a + 60\right)\cdot 89^{7} + \left(25 a^{2} + 74 a + 77\right)\cdot 89^{8} + \left(23 a^{2} + 83 a + 40\right)\cdot 89^{9} + \left(2 a^{2} + 65 a + 87\right)\cdot 89^{10} + \left(47 a^{2} + 80 a + 60\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 a^{2} + 20 a + 12 + \left(82 a^{2} + 22 a + 88\right)\cdot 89 + \left(23 a^{2} + 34 a + 1\right)\cdot 89^{2} + \left(60 a^{2} + 79 a + 17\right)\cdot 89^{3} + \left(4 a^{2} + 62 a + 58\right)\cdot 89^{4} + \left(2 a^{2} + 61 a + 81\right)\cdot 89^{5} + \left(28 a^{2} + 10 a + 40\right)\cdot 89^{6} + \left(62 a^{2} + 49 a + 58\right)\cdot 89^{7} + \left(22 a^{2} + 33 a + 71\right)\cdot 89^{8} + \left(31 a^{2} + 68 a + 56\right)\cdot 89^{9} + \left(49 a^{2} + 5 a + 3\right)\cdot 89^{10} + \left(69 a^{2} + 19 a + 17\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a^{2} + 6 a + 2 + \left(55 a^{2} + 35 a + 32\right)\cdot 89 + \left(37 a^{2} + 51 a + 61\right)\cdot 89^{2} + \left(49 a^{2} + 34 a + 14\right)\cdot 89^{3} + \left(57 a^{2} + 43 a + 13\right)\cdot 89^{4} + \left(84 a^{2} + 37 a + 57\right)\cdot 89^{5} + \left(56 a^{2} + 44 a + 78\right)\cdot 89^{6} + \left(71 a^{2} + 62 a + 66\right)\cdot 89^{7} + \left(18 a^{2} + 86 a + 4\right)\cdot 89^{8} + \left(32 a^{2} + 19 a + 87\right)\cdot 89^{9} + \left(67 a^{2} + 88 a + 88\right)\cdot 89^{10} + \left(86 a^{2} + 4 a + 56\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 77 + 58\cdot 89 + 64\cdot 89^{2} + 59\cdot 89^{3} + 85\cdot 89^{4} + 80\cdot 89^{5} + 34\cdot 89^{6} + 84\cdot 89^{7} + 2\cdot 89^{8} + 25\cdot 89^{9} + 19\cdot 89^{10} + 72\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 9 a^{2} + 7 a + 65 + \left(45 a^{2} + 42 a + 11\right)\cdot 89 + \left(7 a^{2} + 20 a + 1\right)\cdot 89^{2} + \left(10 a^{2} + 86 a + 25\right)\cdot 89^{3} + \left(16 a^{2} + 52 a + 19\right)\cdot 89^{4} + \left(83 a^{2} + 7 a + 54\right)\cdot 89^{5} + \left(42 a^{2} + a + 50\right)\cdot 89^{6} + \left(74 a^{2} + 49 a + 72\right)\cdot 89^{7} + \left(36 a^{2} + 46 a + 40\right)\cdot 89^{8} + \left(43 a^{2} + 86 a + 20\right)\cdot 89^{9} + \left(16 a^{2} + 32 a + 76\right)\cdot 89^{10} + \left(63 a^{2} + 22 a + 9\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 7 + 61\cdot 89 + 72\cdot 89^{2} + 17\cdot 89^{3} + 69\cdot 89^{4} + 43\cdot 89^{5} + 35\cdot 89^{6} + 79\cdot 89^{7} + 56\cdot 89^{8} + 29\cdot 89^{9} + 14\cdot 89^{10} + 20\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 63 a^{2} + 20 a + 28 + \left(15 a^{2} + 62 a + 43\right)\cdot 89 + \left(54 a^{2} + 77 a + 62\right)\cdot 89^{2} + \left(79 a^{2} + 9 a + 55\right)\cdot 89^{3} + \left(29 a^{2} + 22 a + 19\right)\cdot 89^{4} + \left(30 a^{2} + 15 a + 49\right)\cdot 89^{5} + \left(41 a^{2} + 52 a + 67\right)\cdot 89^{6} + \left(52 a^{2} + 76 a + 38\right)\cdot 89^{7} + \left(40 a^{2} + 69 a + 18\right)\cdot 89^{8} + \left(34 a^{2} + 25 a + 63\right)\cdot 89^{9} + \left(37 a^{2} + 17 a + 68\right)\cdot 89^{10} + \left(61 a^{2} + 78 a\right)\cdot 89^{11} +O\left(89^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,8,6)(3,9,4)$ |
| $(1,5,7)(2,8,6)(3,4,9)$ |
| $(1,3,6)(2,5,4)(7,9,8)$ |
| $(2,3)(4,6)(5,7)(8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,8)(2,5)(3,4)(6,7)$ |
$0$ |
| $2$ |
$3$ |
$(1,5,7)(2,8,6)(3,4,9)$ |
$-3$ |
| $3$ |
$3$ |
$(1,7,5)(2,8,6)$ |
$0$ |
| $3$ |
$3$ |
$(1,5,7)(2,6,8)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,6)(2,5,4)(7,9,8)$ |
$0$ |
| $6$ |
$3$ |
$(1,6,3)(2,4,5)(7,8,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,6,4)(2,9,5)(3,7,8)$ |
$0$ |
| $9$ |
$6$ |
$(1,2,7,8,5,6)(3,4)$ |
$0$ |
| $9$ |
$6$ |
$(1,6,5,8,7,2)(3,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
