Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{3} + 3 x + 99 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 + 43\cdot 101 + 63\cdot 101^{2} + 42\cdot 101^{3} + 48\cdot 101^{4} + 74\cdot 101^{5} + 74\cdot 101^{6} + 15\cdot 101^{7} + 13\cdot 101^{8} + 63\cdot 101^{9} + 94\cdot 101^{10} + 11\cdot 101^{11} + 10\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a^{2} + 57 a + 95 + \left(75 a^{2} + 9 a + 43\right)\cdot 101 + \left(69 a^{2} + a + 10\right)\cdot 101^{2} + \left(61 a^{2} + 24 a + 78\right)\cdot 101^{3} + \left(100 a^{2} + 60 a + 17\right)\cdot 101^{4} + \left(a^{2} + 46 a + 55\right)\cdot 101^{5} + \left(12 a^{2} + 31 a + 23\right)\cdot 101^{6} + \left(91 a^{2} + 22 a + 1\right)\cdot 101^{7} + \left(80 a^{2} + 28 a + 95\right)\cdot 101^{8} + \left(34 a^{2} + 86 a + 85\right)\cdot 101^{9} + \left(55 a^{2} + 63 a + 2\right)\cdot 101^{10} + \left(18 a^{2} + 93 a + 54\right)\cdot 101^{11} + \left(46 a^{2} + 79 a + 94\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a^{2} + 22 a + 31 + \left(99 a^{2} + 74 a + 67\right)\cdot 101 + \left(98 a^{2} + 62 a + 27\right)\cdot 101^{2} + \left(14 a^{2} + 79 a + 26\right)\cdot 101^{3} + \left(60 a^{2} + 11 a + 9\right)\cdot 101^{4} + \left(46 a^{2} + 37 a + 95\right)\cdot 101^{5} + \left(70 a^{2} + 38 a + 81\right)\cdot 101^{6} + \left(95 a^{2} + 16 a + 11\right)\cdot 101^{7} + \left(12 a^{2} + 17 a + 37\right)\cdot 101^{8} + \left(59 a^{2} + 25 a + 68\right)\cdot 101^{9} + \left(67 a^{2} + 17 a + 32\right)\cdot 101^{10} + \left(25 a^{2} + 54 a + 69\right)\cdot 101^{11} + \left(64 a^{2} + 36 a + 4\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a^{2} + 45 a + 91 + \left(18 a^{2} + 43 a + 30\right)\cdot 101 + \left(100 a^{2} + 85 a + 71\right)\cdot 101^{2} + \left(27 a^{2} + 48 a + 10\right)\cdot 101^{3} + \left(57 a^{2} + 18 a + 32\right)\cdot 101^{4} + \left(77 a^{2} + 98 a + 4\right)\cdot 101^{5} + \left(5 a^{2} + 62 a + 11\right)\cdot 101^{6} + \left(33 a^{2} + 7 a + 87\right)\cdot 101^{7} + \left(71 a^{2} + 54 a + 75\right)\cdot 101^{8} + \left(15 a^{2} + 25 a + 47\right)\cdot 101^{9} + \left(52 a^{2} + 79 a + 97\right)\cdot 101^{10} + \left(66 a^{2} + 99 a + 48\right)\cdot 101^{11} + \left(31 a^{2} + 73 a + 65\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 78 a^{2} + 51 a + 50 + \left(67 a^{2} + 9 a + 4\right)\cdot 101 + \left(70 a^{2} + 16 a + 72\right)\cdot 101^{2} + \left(78 a^{2} + 36 a + 52\right)\cdot 101^{3} + \left(95 a^{2} + 18 a + 80\right)\cdot 101^{4} + \left(62 a^{2} + 12 a + 26\right)\cdot 101^{5} + \left(76 a^{2} + 3 a + 94\right)\cdot 101^{6} + \left(52 a^{2} + 88 a + 26\right)\cdot 101^{7} + \left(95 a^{2} + 65 a\right)\cdot 101^{8} + \left(41 a^{2} + 56 a + 34\right)\cdot 101^{9} + \left(89 a^{2} + 73 a + 76\right)\cdot 101^{10} + \left(30 a^{2} + 26 a + 79\right)\cdot 101^{11} + \left(77 a^{2} + 33 a + 30\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 + 97\cdot 101 + 34\cdot 101^{2} + 64\cdot 101^{3} + 17\cdot 101^{4} + 31\cdot 101^{5} + 92\cdot 101^{6} + 59\cdot 101^{7} + 83\cdot 101^{8} + 8\cdot 101^{9} + 17\cdot 101^{10} + 80\cdot 101^{11} + 38\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 5 a^{2} + 28 a + 5 + \left(35 a^{2} + 17 a + 40\right)\cdot 101 + \left(32 a^{2} + 22 a + 96\right)\cdot 101^{2} + \left(7 a^{2} + 86 a + 10\right)\cdot 101^{3} + \left(46 a^{2} + 70 a + 82\right)\cdot 101^{4} + \left(92 a^{2} + 51 a + 85\right)\cdot 101^{5} + \left(54 a^{2} + 59 a + 50\right)\cdot 101^{6} + \left(53 a^{2} + 97 a + 28\right)\cdot 101^{7} + \left(93 a^{2} + 17 a + 97\right)\cdot 101^{8} + \left(100 a^{2} + 19 a + 50\right)\cdot 101^{9} + \left(44 a^{2} + 10 a + 88\right)\cdot 101^{10} + \left(44 a^{2} + 20 a + 5\right)\cdot 101^{11} + \left(60 a^{2} + 31 a + 98\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 a^{2} + 100 a + 62 + \left(7 a^{2} + 47 a + 9\right)\cdot 101 + \left(32 a^{2} + 14 a + 36\right)\cdot 101^{2} + \left(11 a^{2} + 28 a + 78\right)\cdot 101^{3} + \left(44 a^{2} + 22 a + 5\right)\cdot 101^{4} + \left(21 a^{2} + 57 a + 94\right)\cdot 101^{5} + \left(83 a^{2} + 6 a + 64\right)\cdot 101^{6} + \left(77 a^{2} + 71 a + 75\right)\cdot 101^{7} + \left(49 a^{2} + 18 a + 32\right)\cdot 101^{8} + \left(50 a^{2} + 90 a + 16\right)\cdot 101^{9} + \left(94 a^{2} + 58 a + 81\right)\cdot 101^{10} + \left(15 a^{2} + 8 a + 48\right)\cdot 101^{11} + \left(23 a^{2} + 48 a + 48\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 15 + 67\cdot 101 + 92\cdot 101^{2} + 39\cdot 101^{3} + 9\cdot 101^{4} + 38\cdot 101^{5} + 11\cdot 101^{6} + 97\cdot 101^{7} + 69\cdot 101^{8} + 28\cdot 101^{9} + 14\cdot 101^{10} + 5\cdot 101^{11} + 13\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,4,3,9,8,7,6,2,5)$ |
| $(1,6,9)(2,8,4)(3,5,7)$ |
| $(2,3)(4,5)(6,9)(7,8)$ |
| $(2,4,8)(3,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(2,3)(4,5)(6,9)(7,8)$ |
$0$ |
| $2$ |
$3$ |
$(1,9,6)(2,4,8)(3,7,5)$ |
$-3$ |
| $3$ |
$3$ |
$(1,6,9)(2,4,8)$ |
$0$ |
| $3$ |
$3$ |
$(1,9,6)(2,8,4)$ |
$0$ |
| $9$ |
$6$ |
$(1,8,6,2,9,4)(3,7)$ |
$0$ |
| $9$ |
$6$ |
$(1,4,9,2,6,8)(3,7)$ |
$0$ |
| $6$ |
$9$ |
$(1,4,3,9,8,7,6,2,5)$ |
$0$ |
| $6$ |
$9$ |
$(1,8,3,9,2,7,6,4,5)$ |
$0$ |
| $6$ |
$9$ |
$(1,3,2,6,5,8,9,7,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
