Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 15.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{3} + 3 x + 99 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 2\cdot 101 + 47\cdot 101^{2} + 16\cdot 101^{3} + 86\cdot 101^{4} + 6\cdot 101^{5} + 66\cdot 101^{6} + 26\cdot 101^{7} + 4\cdot 101^{8} + 45\cdot 101^{9} + 96\cdot 101^{10} + 92\cdot 101^{11} + 4\cdot 101^{12} + 10\cdot 101^{13} + 38\cdot 101^{14} +O\left(101^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 a^{2} + 42 a + 80 + \left(72 a^{2} + 61 a + 97\right)\cdot 101 + \left(67 a^{2} + 65 a + 91\right)\cdot 101^{2} + \left(28 a^{2} + 14 a + 85\right)\cdot 101^{3} + \left(92 a^{2} + 12 a + 39\right)\cdot 101^{4} + \left(57 a^{2} + 69 a + 38\right)\cdot 101^{5} + \left(90 a^{2} + 91 a + 73\right)\cdot 101^{6} + \left(22 a^{2} + 15 a + 29\right)\cdot 101^{7} + \left(36 a^{2} + 46 a + 8\right)\cdot 101^{8} + \left(22 a^{2} + 90 a + 89\right)\cdot 101^{9} + \left(19 a^{2} + 22 a + 76\right)\cdot 101^{10} + \left(83 a^{2} + 47 a + 25\right)\cdot 101^{11} + \left(27 a^{2} + 43 a + 35\right)\cdot 101^{12} + \left(93 a^{2} + 33 a\right)\cdot 101^{13} + \left(36 a^{2} + 23 a + 84\right)\cdot 101^{14} +O\left(101^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 + 36\cdot 101 + 5\cdot 101^{2} + 92\cdot 101^{3} + 50\cdot 101^{4} + 89\cdot 101^{5} + 75\cdot 101^{6} + 85\cdot 101^{7} + 10\cdot 101^{8} + 61\cdot 101^{9} + 6\cdot 101^{10} + 69\cdot 101^{11} + 40\cdot 101^{12} + 78\cdot 101^{13} + 9\cdot 101^{14} +O\left(101^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 73 + 100\cdot 101 + 23\cdot 101^{2} + 23\cdot 101^{3} + 87\cdot 101^{4} + 64\cdot 101^{5} + 91\cdot 101^{6} + 10\cdot 101^{7} + 82\cdot 101^{8} + 61\cdot 101^{9} + 5\cdot 101^{10} + 27\cdot 101^{11} + 47\cdot 101^{12} + 12\cdot 101^{13} + 8\cdot 101^{14} +O\left(101^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 84 a^{2} + 17 a + 65 + \left(44 a^{2} + 60 a + 42\right)\cdot 101 + \left(63 a^{2} + 17 a + 83\right)\cdot 101^{2} + \left(2 a^{2} + 87 a + 33\right)\cdot 101^{3} + \left(69 a^{2} + 59 a + 94\right)\cdot 101^{4} + \left(70 a^{2} + 25 a + 63\right)\cdot 101^{5} + \left(59 a^{2} + 11\right)\cdot 101^{6} + \left(22 a^{2} + 8 a + 29\right)\cdot 101^{7} + \left(71 a^{2} + 35 a + 78\right)\cdot 101^{8} + \left(84 a^{2} + 69 a + 11\right)\cdot 101^{9} + \left(96 a^{2} + 25 a + 30\right)\cdot 101^{10} + \left(34 a^{2} + a + 30\right)\cdot 101^{11} + \left(19 a^{2} + 11 a + 18\right)\cdot 101^{12} + \left(91 a^{2} + 16 a + 97\right)\cdot 101^{13} + \left(88 a^{2} + 70 a + 86\right)\cdot 101^{14} +O\left(101^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 83 a^{2} + 73 a + 49 + \left(97 a^{2} + 81 a + 61\right)\cdot 101 + \left(13 a^{2} + 6 a + 79\right)\cdot 101^{2} + \left(7 a^{2} + 66 a + 42\right)\cdot 101^{3} + \left(51 a^{2} + 67 a + 37\right)\cdot 101^{4} + \left(11 a^{2} + 6 a + 80\right)\cdot 101^{5} + \left(86 a^{2} + 18 a + 33\right)\cdot 101^{6} + \left(55 a^{2} + 14 a + 19\right)\cdot 101^{7} + \left(48 a^{2} + 31 a + 95\right)\cdot 101^{8} + \left(23 a^{2} + 55 a + 13\right)\cdot 101^{9} + \left(46 a + 94\right)\cdot 101^{10} + \left(10 a^{2} + 68 a + 63\right)\cdot 101^{11} + \left(a^{2} + 9 a + 92\right)\cdot 101^{12} + \left(18 a^{2} + 63 a + 53\right)\cdot 101^{13} + \left(71 a^{2} + 39 a + 12\right)\cdot 101^{14} +O\left(101^{ 15 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 77 a^{2} + 42 a + 51 + \left(84 a^{2} + 80 a + 21\right)\cdot 101 + \left(70 a^{2} + 17 a + 98\right)\cdot 101^{2} + \left(69 a^{2} + 100 a + 66\right)\cdot 101^{3} + \left(40 a^{2} + 28 a + 37\right)\cdot 101^{4} + \left(73 a^{2} + 6 a + 69\right)\cdot 101^{5} + \left(51 a^{2} + 9 a + 96\right)\cdot 101^{6} + \left(55 a^{2} + 77 a + 94\right)\cdot 101^{7} + \left(94 a^{2} + 19 a + 23\right)\cdot 101^{8} + \left(94 a^{2} + 42 a + 32\right)\cdot 101^{9} + \left(85 a^{2} + 52 a + 8\right)\cdot 101^{10} + \left(83 a^{2} + 52 a + 27\right)\cdot 101^{11} + \left(53 a^{2} + 46 a + 87\right)\cdot 101^{12} + \left(17 a^{2} + 51 a + 50\right)\cdot 101^{13} + \left(76 a^{2} + 7 a + 61\right)\cdot 101^{14} +O\left(101^{ 15 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 75 a^{2} + 55 a + 33 + \left(92 a^{2} + 5 a + 51\right)\cdot 101 + \left(65 a^{2} + 88 a + 82\right)\cdot 101^{2} + \left(38 a^{2} + 98 a + 4\right)\cdot 101^{3} + \left(10 a^{2} + 34 a + 57\right)\cdot 101^{4} + \left(8 a^{2} + 92 a + 73\right)\cdot 101^{5} + \left(45 a^{2} + 31 a + 52\right)\cdot 101^{6} + \left(57 a^{2} + 4 a + 22\right)\cdot 101^{7} + \left(74 a^{2} + 51 a + 46\right)\cdot 101^{8} + \left(89 a^{2} + 75 a + 45\right)\cdot 101^{9} + \left(8 a^{2} + 61 a + 10\right)\cdot 101^{10} + \left(4 a^{2} + 44 a + 52\right)\cdot 101^{11} + \left(83 a^{2} + 74 a + 54\right)\cdot 101^{12} + \left(66 a^{2} + 89 a + 50\right)\cdot 101^{13} + \left(37 a^{2} + 28 a + 46\right)\cdot 101^{14} +O\left(101^{ 15 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 44 a^{2} + 74 a + 72 + \left(11 a^{2} + 13 a + 90\right)\cdot 101 + \left(21 a^{2} + 6 a + 93\right)\cdot 101^{2} + \left(55 a^{2} + 37 a + 37\right)\cdot 101^{3} + \left(39 a^{2} + 99 a + 14\right)\cdot 101^{4} + \left(81 a^{2} + a + 18\right)\cdot 101^{5} + \left(70 a^{2} + 51 a + 3\right)\cdot 101^{6} + \left(88 a^{2} + 82 a + 85\right)\cdot 101^{7} + \left(78 a^{2} + 18 a + 54\right)\cdot 101^{8} + \left(88 a^{2} + 71 a + 43\right)\cdot 101^{9} + \left(91 a^{2} + 93 a + 75\right)\cdot 101^{10} + \left(86 a^{2} + 88 a + 15\right)\cdot 101^{11} + \left(16 a^{2} + 16 a + 23\right)\cdot 101^{12} + \left(16 a^{2} + 49 a + 50\right)\cdot 101^{13} + \left(93 a^{2} + 32 a + 56\right)\cdot 101^{14} +O\left(101^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(6,9,8)$ |
| $(1,4,3)$ |
| $(1,5)(2,4)(3,7)(6,8,9)$ |
| $(2,5,7)$ |
| $(1,6,4,8,3,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,5)(2,4)(3,7)$ | $1$ |
| $1$ | $3$ | $(1,3,4)(2,5,7)(6,9,8)$ | $-3 \zeta_{3} - 3$ |
| $1$ | $3$ | $(1,4,3)(2,7,5)(6,8,9)$ | $3 \zeta_{3}$ |
| $3$ | $3$ | $(6,9,8)$ | $2 \zeta_{3} + 1$ |
| $3$ | $3$ | $(6,8,9)$ | $-2 \zeta_{3} - 1$ |
| $3$ | $3$ | $(1,4,3)(2,7,5)(6,9,8)$ | $\zeta_{3} + 2$ |
| $3$ | $3$ | $(1,3,4)(2,5,7)(6,8,9)$ | $-\zeta_{3} + 1$ |
| $3$ | $3$ | $(1,4,3)(2,7,5)$ | $-\zeta_{3} - 2$ |
| $3$ | $3$ | $(1,3,4)(2,5,7)$ | $\zeta_{3} - 1$ |
| $6$ | $3$ | $(1,4,3)(2,5,7)$ | $0$ |
| $18$ | $3$ | $(1,5,9)(2,6,4)(3,7,8)$ | $0$ |
| $9$ | $6$ | $(1,5)(2,4)(3,7)(6,8,9)$ | $1$ |
| $9$ | $6$ | $(1,5)(2,4)(3,7)(6,9,8)$ | $1$ |
| $9$ | $6$ | $(1,5,4,2,3,7)(6,8,9)$ | $-\zeta_{3} - 1$ |
| $9$ | $6$ | $(1,7,3,2,4,5)(6,9,8)$ | $\zeta_{3}$ |
| $9$ | $6$ | $(1,5,3,7,4,2)(6,8,9)$ | $\zeta_{3}$ |
| $9$ | $6$ | $(1,2,4,7,3,5)(6,9,8)$ | $-\zeta_{3} - 1$ |
| $9$ | $6$ | $(1,6,4,8,3,9)$ | $-\zeta_{3} - 1$ |
| $9$ | $6$ | $(1,9,3,8,4,6)$ | $\zeta_{3}$ |
| $18$ | $9$ | $(1,5,6,3,7,9,4,2,8)$ | $0$ |
| $18$ | $9$ | $(1,6,7,4,8,5,3,9,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
