Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{3} + 2 x + 98 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 3\cdot 103 + 17\cdot 103^{2} + 3\cdot 103^{3} + 5\cdot 103^{4} + 45\cdot 103^{5} + 51\cdot 103^{6} + 81\cdot 103^{7} + 97\cdot 103^{8} + 34\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 54\cdot 103 + 15\cdot 103^{2} + 40\cdot 103^{3} + 82\cdot 103^{4} + 37\cdot 103^{5} + 49\cdot 103^{6} + 40\cdot 103^{7} + 7\cdot 103^{8} + 53\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 + 45\cdot 103 + 70\cdot 103^{2} + 59\cdot 103^{3} + 15\cdot 103^{4} + 20\cdot 103^{5} + 2\cdot 103^{6} + 84\cdot 103^{7} + 100\cdot 103^{8} + 14\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a^{2} + 70 a + 60 + \left(55 a^{2} + 19 a + 73\right)\cdot 103 + \left(98 a^{2} + 8 a + 62\right)\cdot 103^{2} + \left(24 a^{2} + 50 a + 67\right)\cdot 103^{3} + \left(38 a^{2} + 57 a + 16\right)\cdot 103^{4} + \left(73 a^{2} + 68 a + 29\right)\cdot 103^{5} + \left(101 a^{2} + 4 a + 101\right)\cdot 103^{6} + \left(15 a^{2} + 55 a + 89\right)\cdot 103^{7} + \left(93 a^{2} + 26 a + 89\right)\cdot 103^{8} + \left(76 a^{2} + 69 a + 33\right)\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a^{2} + 95 a + 35 + \left(101 a^{2} + 79 a + 66\right)\cdot 103 + \left(95 a^{2} + 57 a + 93\right)\cdot 103^{2} + \left(88 a^{2} + 73 a + 49\right)\cdot 103^{3} + \left(3 a^{2} + 77 a + 39\right)\cdot 103^{4} + \left(61 a^{2} + 102 a + 81\right)\cdot 103^{5} + \left(38 a^{2} + 102 a + 85\right)\cdot 103^{6} + \left(12 a^{2} + 3 a + 50\right)\cdot 103^{7} + \left(32 a^{2} + 24 a + 8\right)\cdot 103^{8} + \left(93 a^{2} + 76 a + 90\right)\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 a^{2} + 29 a + 1 + \left(25 a^{2} + 46 a + 34\right)\cdot 103 + \left(25 a^{2} + 18 a + 102\right)\cdot 103^{2} + \left(57 a^{2} + 93 a + 41\right)\cdot 103^{3} + \left(18 a^{2} + 100 a + 93\right)\cdot 103^{4} + \left(58 a^{2} + 100 a + 8\right)\cdot 103^{5} + \left(82 a^{2} + 3 a + 7\right)\cdot 103^{6} + \left(6 a^{2} + 72 a + 9\right)\cdot 103^{7} + \left(40 a^{2} + 86 a + 19\right)\cdot 103^{8} + \left(49 a^{2} + 95 a + 100\right)\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 58 a^{2} + 41 a + 9 + \left(49 a^{2} + 3 a + 66\right)\cdot 103 + \left(11 a^{2} + 37 a + 49\right)\cdot 103^{2} + \left(92 a^{2} + 82 a + 88\right)\cdot 103^{3} + \left(60 a^{2} + 70 a + 46\right)\cdot 103^{4} + \left(71 a^{2} + 34 a + 95\right)\cdot 103^{5} + \left(65 a^{2} + 98 a + 18\right)\cdot 103^{6} + \left(74 a^{2} + 43 a + 65\right)\cdot 103^{7} + \left(80 a^{2} + 52 a + 4\right)\cdot 103^{8} + \left(35 a^{2} + 60 a + 82\right)\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 74 a^{2} + 5 a + 99 + \left(86 a^{2} + 71 a + 46\right)\cdot 103 + \left(18 a^{2} + 80 a + 59\right)\cdot 103^{2} + \left(100 a^{2} + 88 a + 30\right)\cdot 103^{3} + \left(68 a^{2} + 96 a + 23\right)\cdot 103^{4} + \left(87 a^{2} + 30 a + 48\right)\cdot 103^{5} + \left(87 a^{2} + 29 a + 48\right)\cdot 103^{6} + \left(45 a^{2} + 91 a + 95\right)\cdot 103^{7} + \left(40 a^{2} + 45 a + 53\right)\cdot 103^{8} + \left(70 a^{2} + 48 a + 59\right)\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 80 a^{2} + 69 a + 4 + \left(93 a^{2} + 88 a + 22\right)\cdot 103 + \left(58 a^{2} + 3 a + 44\right)\cdot 103^{2} + \left(48 a^{2} + 24 a + 30\right)\cdot 103^{3} + \left(15 a^{2} + 8 a + 89\right)\cdot 103^{4} + \left(60 a^{2} + 74 a + 45\right)\cdot 103^{5} + \left(35 a^{2} + 69 a + 47\right)\cdot 103^{6} + \left(50 a^{2} + 42 a + 101\right)\cdot 103^{7} + \left(22 a^{2} + 73 a + 29\right)\cdot 103^{8} + \left(86 a^{2} + 61 a + 46\right)\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,7,5)(6,9,8)$ |
| $(1,3,2)(4,5,7)(6,9,8)$ |
| $(4,6)(5,9)(7,8)$ |
| $(1,6,4)(2,8,7)(3,9,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(4,6)(5,9)(7,8)$ | $1$ |
| $1$ | $3$ | $(1,3,2)(4,5,7)(6,9,8)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,2,3)(4,7,5)(6,8,9)$ | $-3 \zeta_{3} - 3$ |
| $6$ | $3$ | $(1,6,4)(2,8,7)(3,9,5)$ | $0$ |
| $6$ | $3$ | $(1,9,4)(2,6,7)(3,8,5)$ | $0$ |
| $6$ | $3$ | $(1,8,4)(2,9,7)(3,6,5)$ | $0$ |
| $6$ | $3$ | $(4,7,5)(6,9,8)$ | $0$ |
| $9$ | $6$ | $(1,3,2)(4,9,7,6,5,8)$ | $\zeta_{3}$ |
| $9$ | $6$ | $(1,2,3)(4,8,5,6,7,9)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
