Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{3} + 2 x + 98 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 + 82\cdot 103 + 86\cdot 103^{2} + 79\cdot 103^{3} + 33\cdot 103^{4} + 91\cdot 103^{5} + 45\cdot 103^{6} + 77\cdot 103^{7} + 13\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 69 + 85\cdot 103 + 90\cdot 103^{2} + 78\cdot 103^{3} + 37\cdot 103^{4} + 17\cdot 103^{5} + 10\cdot 103^{6} + 79\cdot 103^{7} + 35\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 84 + 37\cdot 103 + 28\cdot 103^{2} + 47\cdot 103^{3} + 31\cdot 103^{4} + 97\cdot 103^{5} + 46\cdot 103^{6} + 49\cdot 103^{7} + 53\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a^{2} + 55 a + 37 + \left(32 a^{2} + 75 a + 8\right)\cdot 103 + \left(64 a^{2} + 101 a + 17\right)\cdot 103^{2} + \left(75 a^{2} + 58 a + 32\right)\cdot 103^{3} + \left(94 a^{2} + 40 a + 23\right)\cdot 103^{4} + \left(33 a^{2} + 74 a + 45\right)\cdot 103^{5} + \left(98 a^{2} + 52 a + 62\right)\cdot 103^{6} + \left(43 a^{2} + 46 a + 58\right)\cdot 103^{7} + \left(45 a^{2} + 7 a + 60\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a^{2} + 3 a + 41 + \left(73 a^{2} + 29 a + 97\right)\cdot 103 + \left(19 a^{2} + 73 a + 94\right)\cdot 103^{2} + \left(102 a^{2} + 20 a + 101\right)\cdot 103^{3} + \left(48 a^{2} + 63 a + 30\right)\cdot 103^{4} + \left(55 a^{2} + 60 a + 5\right)\cdot 103^{5} + \left(20 a^{2} + 93 a + 96\right)\cdot 103^{6} + \left(29 a^{2} + 24 a + 38\right)\cdot 103^{7} + \left(13 a^{2} + 93 a + 86\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a^{2} + 93 a + 12 + \left(77 a^{2} + 93 a + 34\right)\cdot 103 + \left(84 a^{2} + 12 a + 44\right)\cdot 103^{2} + \left(16 a^{2} + 14 a + 22\right)\cdot 103^{3} + \left(60 a^{2} + 37 a + 80\right)\cdot 103^{4} + \left(84 a^{2} + 71 a + 9\right)\cdot 103^{5} + \left(31 a^{2} + 51 a + 8\right)\cdot 103^{6} + \left(18 a^{2} + 76 a + 93\right)\cdot 103^{7} + \left(15 a^{2} + 59 a + 88\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 24 a^{2} + 35 a + 32 + \left(23 a^{2} + 95 a + 65\right)\cdot 103 + \left(14 a^{2} + 39 a + 87\right)\cdot 103^{2} + \left(78 a^{2} + 38 a + 69\right)\cdot 103^{3} + \left(69 a^{2} + 41 a + 58\right)\cdot 103^{4} + \left(82 a^{2} + 68 a + 41\right)\cdot 103^{5} + \left(26 a^{2} + 64 a + 1\right)\cdot 103^{6} + \left(53 a^{2} + 59 a + 71\right)\cdot 103^{7} + \left(7 a^{2} + 13 a + 78\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 74 a^{2} + 65 a + 30 + \left(6 a^{2} + 81 a + 43\right)\cdot 103 + \left(69 a^{2} + 92 a + 23\right)\cdot 103^{2} + \left(25 a^{2} + 43 a + 34\right)\cdot 103^{3} + \left(87 a^{2} + 101 a + 13\right)\cdot 103^{4} + \left(67 a^{2} + 76 a + 56\right)\cdot 103^{5} + \left(55 a^{2} + 47 a + 5\right)\cdot 103^{6} + \left(20 a^{2} + 18 a + 96\right)\cdot 103^{7} + \left(82 a^{2} + 99 a + 40\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 92 a^{2} + 58 a + 54 + \left(96 a^{2} + 36 a + 60\right)\cdot 103 + \left(56 a^{2} + 91 a + 41\right)\cdot 103^{2} + \left(10 a^{2} + 29 a + 48\right)\cdot 103^{3} + \left(51 a^{2} + 25 a + 102\right)\cdot 103^{4} + \left(87 a^{2} + 60 a + 47\right)\cdot 103^{5} + \left(75 a^{2} + 101 a + 32\right)\cdot 103^{6} + \left(40 a^{2} + 82 a + 54\right)\cdot 103^{7} + \left(42 a^{2} + 35 a + 56\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,7)(5,6)(8,9)$ |
| $(1,2)(4,6)(5,7)$ |
| $(1,4,7)(2,9,8)(3,6,5)$ |
| $(4,6,9)(5,7,8)$ |
| $(1,3,2)(4,6,9)(5,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,2)(6,9)(7,8)$ |
$0$ |
| $9$ |
$2$ |
$(4,7)(5,6)(8,9)$ |
$-2$ |
| $9$ |
$2$ |
$(1,4)(2,6)(3,9)(5,7)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,2)(4,6,9)(5,8,7)$ |
$-3$ |
| $6$ |
$3$ |
$(1,7,9)(2,8,6)(3,5,4)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,2)(4,9,6)$ |
$0$ |
| $12$ |
$3$ |
$(1,4,7)(2,9,8)(3,6,5)$ |
$0$ |
| $18$ |
$6$ |
$(1,6,7,2,9,8)(3,4,5)$ |
$0$ |
| $18$ |
$6$ |
$(1,3,2)(4,5,9,7,6,8)$ |
$1$ |
| $18$ |
$6$ |
$(1,6,3,4,2,9)(5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
