Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{3} + 3 x + 99 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 60 a^{2} + 95 a + 85 + \left(82 a^{2} + 28 a + 74\right)\cdot 101 + \left(a^{2} + 49 a + 92\right)\cdot 101^{2} + \left(45 a^{2} + 61 a + 82\right)\cdot 101^{3} + \left(100 a^{2} + 92 a + 100\right)\cdot 101^{4} + \left(59 a^{2} + 75 a\right)\cdot 101^{5} + \left(43 a^{2} + 89 a + 76\right)\cdot 101^{6} + \left(27 a^{2} + 87 a + 96\right)\cdot 101^{7} + \left(61 a^{2} + 79 a + 52\right)\cdot 101^{8} + \left(66 a^{2} + 57 a + 15\right)\cdot 101^{9} + \left(94 a^{2} + 68 a + 77\right)\cdot 101^{10} + \left(83 a^{2} + 29 a + 76\right)\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 + 72\cdot 101 + 12\cdot 101^{2} + 46\cdot 101^{3} + 22\cdot 101^{4} + 11\cdot 101^{5} + 52\cdot 101^{6} + 90\cdot 101^{7} + 92\cdot 101^{8} + 64\cdot 101^{9} + 94\cdot 101^{10} + 76\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 98 a^{2} + 90 a + 18 + \left(47 a^{2} + 57 a + 46\right)\cdot 101 + \left(54 a^{2} + 2 a + 84\right)\cdot 101^{2} + \left(99 a^{2} + 18 a + 72\right)\cdot 101^{3} + \left(10 a^{2} + 38 a + 86\right)\cdot 101^{4} + \left(71 a^{2} + 28 a + 98\right)\cdot 101^{5} + \left(74 a^{2} + 9 a + 40\right)\cdot 101^{6} + \left(31 a^{2} + 71 a + 50\right)\cdot 101^{7} + \left(48 a^{2} + 20 a + 73\right)\cdot 101^{8} + \left(64 a^{2} + 11 a + 6\right)\cdot 101^{9} + \left(43 a^{2} + 60 a + 68\right)\cdot 101^{10} + \left(54 a^{2} + 13 a + 92\right)\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 80 a^{2} + 4 a + 83 + \left(50 a^{2} + 73 a + 51\right)\cdot 101 + \left(63 a^{2} + 49 a + 1\right)\cdot 101^{2} + \left(13 a^{2} + 32 a + 2\right)\cdot 101^{3} + \left(61 a^{2} + 37 a + 86\right)\cdot 101^{4} + \left(87 a^{2} + 56 a + 30\right)\cdot 101^{5} + \left(7 a^{2} + 15 a + 8\right)\cdot 101^{6} + \left(87 a^{2} + 26 a + 60\right)\cdot 101^{7} + \left(42 a^{2} + 31 a + 62\right)\cdot 101^{8} + \left(49 a^{2} + 78 a + 77\right)\cdot 101^{9} + \left(33 a^{2} + 81 a + 47\right)\cdot 101^{10} + \left(18 a^{2} + 14 a + 20\right)\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 + 98\cdot 101 + 81\cdot 101^{2} + 40\cdot 101^{3} + 28\cdot 101^{4} + 32\cdot 101^{5} + 14\cdot 101^{6} + 28\cdot 101^{7} + 80\cdot 101^{8} + 54\cdot 101^{9} + 90\cdot 101^{10} + 65\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a^{2} + 7 a + 72 + \left(2 a^{2} + 71 a + 55\right)\cdot 101 + \left(84 a^{2} + 48 a + 42\right)\cdot 101^{2} + \left(88 a^{2} + 50 a + 51\right)\cdot 101^{3} + \left(28 a^{2} + 25 a + 21\right)\cdot 101^{4} + \left(43 a^{2} + 16 a + 43\right)\cdot 101^{5} + \left(18 a^{2} + 76 a + 29\right)\cdot 101^{6} + \left(83 a^{2} + 3 a + 52\right)\cdot 101^{7} + \left(9 a^{2} + 49 a + 97\right)\cdot 101^{8} + \left(88 a^{2} + 11 a + 53\right)\cdot 101^{9} + \left(23 a^{2} + 60 a + 28\right)\cdot 101^{10} + \left(28 a^{2} + 72 a + 40\right)\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 100 a^{2} + 16 a + 64 + \left(61 a^{2} + 36 a + 33\right)\cdot 101 + \left(97 a^{2} + 40 a + 82\right)\cdot 101^{2} + \left(40 a^{2} + 83 a + 74\right)\cdot 101^{3} + \left(a^{2} + 23 a + 3\right)\cdot 101^{4} + \left(88 a^{2} + 81 a + 57\right)\cdot 101^{5} + \left(14 a^{2} + 57 a + 18\right)\cdot 101^{6} + \left(16 a^{2} + 54 a + 74\right)\cdot 101^{7} + \left(89 a^{2} + 23 a + 7\right)\cdot 101^{8} + \left(3 a^{2} + 89 a + 92\right)\cdot 101^{9} + \left(3 a^{2} + 74 a + 95\right)\cdot 101^{10} + \left(87 a^{2} + 37 a + 82\right)\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 29 + 47\cdot 101 + 14\cdot 101^{2} + 10\cdot 101^{3} + 55\cdot 101^{4} + 39\cdot 101^{5} + 90\cdot 101^{6} + 97\cdot 101^{7} + 3\cdot 101^{8} + 95\cdot 101^{9} + 6\cdot 101^{10} + 78\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 42 a^{2} + 91 a + 49 + \left(57 a^{2} + 35 a + 24\right)\cdot 101 + \left(a^{2} + 11 a + 92\right)\cdot 101^{2} + \left(15 a^{2} + 57 a + 22\right)\cdot 101^{3} + \left(100 a^{2} + 85 a + 100\right)\cdot 101^{4} + \left(53 a^{2} + 44 a + 89\right)\cdot 101^{5} + \left(42 a^{2} + 54 a + 73\right)\cdot 101^{6} + \left(57 a^{2} + 59 a + 55\right)\cdot 101^{7} + \left(51 a^{2} + 98 a + 33\right)\cdot 101^{8} + \left(30 a^{2} + 54 a + 44\right)\cdot 101^{9} + \left(3 a^{2} + 58 a + 96\right)\cdot 101^{10} + \left(31 a^{2} + 33 a + 71\right)\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,5)(3,4)(7,9)$ |
| $(1,9,7)(3,6,4)$ |
| $(1,6,8)(2,7,4)(3,5,9)$ |
| $(1,7,9)(2,5,8)(3,6,4)$ |
| $(1,6)(3,9)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(2,5)(3,4)(7,9)$ |
$0$ |
| $9$ |
$2$ |
$(1,6)(3,9)(4,7)$ |
$2$ |
| $9$ |
$2$ |
$(1,6)(2,5)(3,7)(4,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,7,9)(2,5,8)(3,6,4)$ |
$-3$ |
| $6$ |
$3$ |
$(1,6,8)(2,7,4)(3,5,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,9,7)(2,5,8)$ |
$0$ |
| $12$ |
$3$ |
$(1,4,2)(3,5,7)(6,8,9)$ |
$0$ |
| $18$ |
$6$ |
$(1,6,8)(2,9,4,5,7,3)$ |
$0$ |
| $18$ |
$6$ |
$(1,4,9,6,7,3)(2,5,8)$ |
$-1$ |
| $18$ |
$6$ |
$(1,8,9,2,7,5)(4,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
