Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 17.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{3} + 9 x + 92 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 89 a^{2} + 50 a + 49 + \left(83 a^{2} + 91 a + 18\right)\cdot 97 + \left(22 a^{2} + 5 a + 40\right)\cdot 97^{2} + \left(90 a^{2} + 18 a + 56\right)\cdot 97^{3} + \left(5 a^{2} + 22 a + 35\right)\cdot 97^{4} + \left(16 a^{2} + 3 a + 96\right)\cdot 97^{5} + \left(62 a^{2} + 69 a + 81\right)\cdot 97^{6} + \left(43 a^{2} + 92 a + 67\right)\cdot 97^{7} + \left(64 a^{2} + 57 a + 95\right)\cdot 97^{8} + \left(27 a^{2} + 85 a + 68\right)\cdot 97^{9} + \left(2 a^{2} + 80 a + 13\right)\cdot 97^{10} + \left(90 a^{2} + 66 a + 55\right)\cdot 97^{11} + \left(58 a^{2} + 27 a + 62\right)\cdot 97^{12} + \left(11 a^{2} + 25 a + 69\right)\cdot 97^{13} + \left(87 a^{2} + 84 a + 37\right)\cdot 97^{14} + \left(76 a^{2} + 43 a + 73\right)\cdot 97^{15} + \left(95 a^{2} + 69 a + 89\right)\cdot 97^{16} +O\left(97^{ 17 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a^{2} + 15 a + 42 + \left(52 a^{2} + 3 a + 21\right)\cdot 97 + \left(47 a^{2} + 95 a + 91\right)\cdot 97^{2} + \left(39 a^{2} + 81 a + 42\right)\cdot 97^{3} + \left(11 a^{2} + 73 a + 68\right)\cdot 97^{4} + \left(a^{2} + 47 a + 6\right)\cdot 97^{5} + \left(87 a^{2} + 37\right)\cdot 97^{6} + \left(37 a^{2} + 90 a + 33\right)\cdot 97^{7} + \left(50 a^{2} + 9 a + 11\right)\cdot 97^{8} + \left(63 a^{2} + 96 a + 90\right)\cdot 97^{9} + \left(8 a^{2} + 78 a + 51\right)\cdot 97^{10} + \left(77 a^{2} + 51 a + 74\right)\cdot 97^{11} + \left(7 a^{2} + 44 a + 46\right)\cdot 97^{12} + \left(27 a^{2} + 48 a + 65\right)\cdot 97^{13} + \left(39 a^{2} + 68 a + 41\right)\cdot 97^{14} + \left(72 a^{2} + 42 a + 46\right)\cdot 97^{15} + \left(93 a^{2} + 26 a + 77\right)\cdot 97^{16} +O\left(97^{ 17 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 + 76\cdot 97 + 33\cdot 97^{2} + 37\cdot 97^{3} + 13\cdot 97^{4} + 49\cdot 97^{5} + 50\cdot 97^{6} + 34\cdot 97^{7} + 10\cdot 97^{8} + 39\cdot 97^{9} + 12\cdot 97^{10} + 87\cdot 97^{11} + 48\cdot 97^{12} + 22\cdot 97^{13} + 12\cdot 97^{14} + 88\cdot 97^{15} + 27\cdot 97^{16} +O\left(97^{ 17 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a^{2} + 4 a + 66 + \left(23 a^{2} + 27 a + 41\right)\cdot 97 + \left(21 a^{2} + 74 a + 30\right)\cdot 97^{2} + \left(a^{2} + 16 a + 7\right)\cdot 97^{3} + \left(88 a^{2} + 88 a + 43\right)\cdot 97^{4} + \left(62 a^{2} + 30 a + 86\right)\cdot 97^{5} + \left(81 a^{2} + 96 a + 4\right)\cdot 97^{6} + \left(51 a^{2} + 56 a + 20\right)\cdot 97^{7} + \left(35 a^{2} + 96 a + 19\right)\cdot 97^{8} + \left(57 a^{2} + 32 a + 53\right)\cdot 97^{9} + \left(54 a^{2} + 65 a + 36\right)\cdot 97^{10} + \left(29 a^{2} + 96 a + 80\right)\cdot 97^{11} + \left(69 a^{2} + 14 a + 27\right)\cdot 97^{12} + \left(74 a^{2} + 81 a + 60\right)\cdot 97^{13} + \left(46 a^{2} + 72 a + 86\right)\cdot 97^{14} + \left(93 a^{2} + 55 a + 75\right)\cdot 97^{15} + \left(34 a^{2} + 2 a + 15\right)\cdot 97^{16} +O\left(97^{ 17 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 a^{2} + 40 a + 15 + \left(31 a^{2} + 69 a + 92\right)\cdot 97 + \left(92 a^{2} + 77 a + 68\right)\cdot 97^{2} + \left(3 a^{2} + 90 a + 23\right)\cdot 97^{3} + \left(17 a^{2} + 56 a + 5\right)\cdot 97^{4} + \left(94 a^{2} + 61 a + 80\right)\cdot 97^{5} + \left(40 a^{2} + 6 a + 51\right)\cdot 97^{6} + \left(26 a^{2} + 13 a + 61\right)\cdot 97^{7} + \left(48 a^{2} + 22 a + 95\right)\cdot 97^{8} + \left(54 a^{2} + 68 a + 35\right)\cdot 97^{9} + \left(45 a^{2} + 81 a + 79\right)\cdot 97^{10} + \left(54 a^{2} + 40 a + 35\right)\cdot 97^{11} + \left(92 a^{2} + 60 a + 70\right)\cdot 97^{12} + \left(87 a^{2} + 11 a + 42\right)\cdot 97^{13} + \left(78 a^{2} + 74 a + 85\right)\cdot 97^{14} + \left(15 a^{2} + 36 a + 94\right)\cdot 97^{15} + \left(34 a^{2} + 72 a + 10\right)\cdot 97^{16} +O\left(97^{ 17 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 14\cdot 97 + 9\cdot 97^{2} + 41\cdot 97^{3} + 27\cdot 97^{4} + 41\cdot 97^{5} + 42\cdot 97^{6} + 40\cdot 97^{7} + 30\cdot 97^{9} + 76\cdot 97^{10} + 12\cdot 97^{11} + 14\cdot 97^{13} + 12\cdot 97^{14} + 28\cdot 97^{15} + 40\cdot 97^{16} +O\left(97^{ 17 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 94 a^{2} + 43 a + 79 + \left(86 a^{2} + 75 a + 36\right)\cdot 97 + \left(52 a^{2} + 16 a + 26\right)\cdot 97^{2} + \left(5 a^{2} + 62 a + 33\right)\cdot 97^{3} + \left(3 a^{2} + 83 a + 18\right)\cdot 97^{4} + \left(18 a^{2} + 62 a + 11\right)\cdot 97^{5} + \left(50 a^{2} + 28 a + 10\right)\cdot 97^{6} + \left(a^{2} + 44 a + 9\right)\cdot 97^{7} + \left(94 a^{2} + 39 a + 79\right)\cdot 97^{8} + \left(11 a^{2} + 75 a + 71\right)\cdot 97^{9} + \left(40 a^{2} + 47 a + 46\right)\cdot 97^{10} + \left(74 a^{2} + 30 a + 58\right)\cdot 97^{11} + \left(65 a^{2} + 54 a + 6\right)\cdot 97^{12} + \left(10 a^{2} + 87 a + 64\right)\cdot 97^{13} + \left(60 a^{2} + 36 a + 69\right)\cdot 97^{14} + \left(23 a^{2} + 94 a + 44\right)\cdot 97^{15} + \left(63 a^{2} + 24 a + 88\right)\cdot 97^{16} +O\left(97^{ 17 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 39 a^{2} + 42 a + 40 + \left(13 a^{2} + 24 a + 80\right)\cdot 97 + \left(54 a^{2} + 21 a + 33\right)\cdot 97^{2} + \left(53 a^{2} + 21 a + 30\right)\cdot 97^{3} + \left(68 a^{2} + 63 a + 23\right)\cdot 97^{4} + \left(a^{2} + 84 a + 10\right)\cdot 97^{5} + \left(66 a^{2} + 89 a + 8\right)\cdot 97^{6} + \left(32 a^{2} + 90 a + 2\right)\cdot 97^{7} + \left(95 a^{2} + 64 a + 87\right)\cdot 97^{8} + \left(75 a^{2} + 29 a + 67\right)\cdot 97^{9} + \left(42 a^{2} + 33 a + 62\right)\cdot 97^{10} + \left(62 a^{2} + 4 a + 83\right)\cdot 97^{11} + \left(93 a^{2} + 89 a + 76\right)\cdot 97^{12} + \left(78 a^{2} + 36 a + 85\right)\cdot 97^{13} + \left(75 a^{2} + 51 a + 66\right)\cdot 97^{14} + \left(8 a^{2} + 17 a + 52\right)\cdot 97^{15} + \left(66 a^{2} + 95 a + 8\right)\cdot 97^{16} +O\left(97^{ 17 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 55 + 5\cdot 97 + 54\cdot 97^{2} + 18\cdot 97^{3} + 56\cdot 97^{4} + 6\cdot 97^{5} + 4\cdot 97^{6} + 22\cdot 97^{7} + 86\cdot 97^{8} + 27\cdot 97^{9} + 8\cdot 97^{10} + 94\cdot 97^{11} + 47\cdot 97^{12} + 60\cdot 97^{13} + 72\cdot 97^{14} + 77\cdot 97^{15} + 28\cdot 97^{16} +O\left(97^{ 17 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,5)(2,7,9)(4,6,8)$ |
| $(2,9)(3,8)(4,7)(5,6)$ |
| $(1,4,7)(2,5,8)(3,6,9)$ |
| $(2,5,8)(3,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $2$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ | $-3$ |
| $3$ | $3$ | $(1,7,4)(2,5,8)$ | $0$ |
| $3$ | $3$ | $(1,4,7)(2,8,5)$ | $0$ |
| $6$ | $3$ | $(1,3,5)(2,7,9)(4,6,8)$ | $0$ |
| $6$ | $3$ | $(1,5,3)(2,9,7)(4,8,6)$ | $0$ |
| $6$ | $3$ | $(1,5,6)(2,3,7)(4,8,9)$ | $0$ |
| $9$ | $6$ | $(1,8,7,2,4,5)(3,6)$ | $0$ |
| $9$ | $6$ | $(1,5,4,2,7,8)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
