Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{3} + 2 x + 98 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 + 82\cdot 103 + 11\cdot 103^{2} + 45\cdot 103^{3} + 2\cdot 103^{4} + 66\cdot 103^{5} + 77\cdot 103^{6} + 54\cdot 103^{7} + 11\cdot 103^{8} + 6\cdot 103^{9} + 88\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 71 + 52\cdot 103 + 84\cdot 103^{2} + 59\cdot 103^{3} + 23\cdot 103^{4} + 21\cdot 103^{5} + 50\cdot 103^{6} + 97\cdot 103^{7} + 99\cdot 103^{8} + 63\cdot 103^{9} + 81\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 86 + 70\cdot 103 + 6\cdot 103^{2} + 101\cdot 103^{3} + 76\cdot 103^{4} + 15\cdot 103^{5} + 78\cdot 103^{6} + 53\cdot 103^{7} + 94\cdot 103^{8} + 32\cdot 103^{9} + 36\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a^{2} + 94 a + 8 + \left(70 a^{2} + 50 a + 59\right)\cdot 103 + \left(14 a^{2} + 69 a + 19\right)\cdot 103^{2} + \left(10 a^{2} + 16 a + 82\right)\cdot 103^{3} + \left(97 a^{2} + 60\right)\cdot 103^{4} + \left(47 a^{2} + 29\right)\cdot 103^{5} + \left(32 a^{2} + 12 a + 43\right)\cdot 103^{6} + \left(12 a^{2} + 99 a + 16\right)\cdot 103^{7} + \left(84 a^{2} + 49 a + 9\right)\cdot 103^{8} + \left(94 a^{2} + 19 a + 92\right)\cdot 103^{9} + \left(41 a^{2} + 13 a + 55\right)\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a^{2} + 13 a + 20 + \left(99 a^{2} + 67 a + 29\right)\cdot 103 + \left(46 a^{2} + 69 a + 28\right)\cdot 103^{2} + \left(42 a^{2} + 23 a + 22\right)\cdot 103^{3} + \left(58 a^{2} + 10 a + 9\right)\cdot 103^{4} + \left(41 a^{2} + 70 a + 21\right)\cdot 103^{5} + \left(26 a^{2} + 2 a + 35\right)\cdot 103^{6} + \left(22 a^{2} + 90 a + 98\right)\cdot 103^{7} + \left(5 a^{2} + 55 a + 6\right)\cdot 103^{8} + \left(5 a^{2} + 47 a + 41\right)\cdot 103^{9} + \left(70 a^{2} + 23 a + 93\right)\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a^{2} + 76 a + 69 + \left(59 a^{2} + 24 a + 44\right)\cdot 103 + \left(83 a^{2} + 80 a + 8\right)\cdot 103^{2} + \left(53 a^{2} + 13 a + 3\right)\cdot 103^{3} + \left(32 a^{2} + 74 a + 9\right)\cdot 103^{4} + \left(89 a^{2} + 102 a + 16\right)\cdot 103^{5} + \left(59 a^{2} + 8 a + 11\right)\cdot 103^{6} + \left(25 a^{2} + 34 a + 34\right)\cdot 103^{7} + \left(48 a^{2} + 26 a + 64\right)\cdot 103^{8} + \left(6 a^{2} + 43 a + 8\right)\cdot 103^{9} + \left(53 a^{2} + 53 a + 2\right)\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 a^{2} + 46 a + 97 + \left(69 a^{2} + 3 a + 23\right)\cdot 103 + \left(41 a^{2} + 95 a + 21\right)\cdot 103^{2} + \left(77 a^{2} + 11 a\right)\cdot 103^{3} + \left(80 a^{2} + 55 a + 39\right)\cdot 103^{4} + \left(48 a^{2} + 89 a + 99\right)\cdot 103^{5} + \left(62 a^{2} + 69 a + 48\right)\cdot 103^{6} + \left(57 a^{2} + a + 42\right)\cdot 103^{7} + \left(24 a^{2} + 83 a + 101\right)\cdot 103^{8} + \left(58 a^{2} + 102 a + 8\right)\cdot 103^{9} + \left(76 a^{2} + 21 a + 102\right)\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 50 a^{2} + 66 a + 101 + \left(66 a^{2} + 48 a + 19\right)\cdot 103 + \left(46 a^{2} + 41 a + 62\right)\cdot 103^{2} + \left(15 a^{2} + 74 a + 20\right)\cdot 103^{3} + \left(28 a^{2} + 47 a + 3\right)\cdot 103^{4} + \left(6 a^{2} + 13 a + 77\right)\cdot 103^{5} + \left(8 a^{2} + 21 a + 10\right)\cdot 103^{6} + \left(33 a^{2} + 2 a + 44\right)\cdot 103^{7} + \left(97 a^{2} + 73 a + 95\right)\cdot 103^{8} + \left(52 a^{2} + 83 a + 1\right)\cdot 103^{9} + \left(87 a^{2} + 67 a + 48\right)\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 62 a^{2} + 14 a + 14 + \left(47 a^{2} + 11 a + 29\right)\cdot 103 + \left(75 a^{2} + 56 a + 66\right)\cdot 103^{2} + \left(6 a^{2} + 65 a + 77\right)\cdot 103^{3} + \left(12 a^{2} + 18 a + 84\right)\cdot 103^{4} + \left(75 a^{2} + 33 a + 65\right)\cdot 103^{5} + \left(16 a^{2} + 91 a + 56\right)\cdot 103^{6} + \left(55 a^{2} + 81 a + 73\right)\cdot 103^{7} + \left(49 a^{2} + 20 a + 31\right)\cdot 103^{8} + \left(91 a^{2} + 12 a + 53\right)\cdot 103^{9} + \left(82 a^{2} + 26 a + 7\right)\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2)(5,9,6)$ |
| $(1,6,4)(2,5,7)(3,9,8)$ |
| $(2,3)(4,7)(5,6)$ |
| $(4,6)(5,7)(8,9)$ |
| $(4,8,7)(5,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(2,3)(6,9)(7,8)$ |
$0$ |
| $9$ |
$2$ |
$(4,6)(5,7)(8,9)$ |
$-2$ |
| $9$ |
$2$ |
$(1,9)(2,5)(3,6)(4,7)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,3)(4,7,8)(5,9,6)$ |
$-3$ |
| $6$ |
$3$ |
$(1,3,2)(5,9,6)$ |
$0$ |
| $6$ |
$3$ |
$(1,4,5)(2,7,9)(3,8,6)$ |
$0$ |
| $12$ |
$3$ |
$(1,6,4)(2,5,7)(3,9,8)$ |
$0$ |
| $18$ |
$6$ |
$(1,5,4)(2,6,7,3,9,8)$ |
$0$ |
| $18$ |
$6$ |
$(1,3,2)(4,5,7,9,8,6)$ |
$1$ |
| $18$ |
$6$ |
$(1,5,3,9,2,6)(4,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
