Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 18.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{3} + 9 x + 92 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 46 a^{2} + 57 a + 82 + \left(81 a^{2} + 89 a + 3\right)\cdot 97 + \left(91 a^{2} + 84 a + 66\right)\cdot 97^{2} + \left(95 a^{2} + 55 a + 90\right)\cdot 97^{3} + \left(80 a^{2} + 48 a\right)\cdot 97^{4} + \left(27 a^{2} + 42 a + 70\right)\cdot 97^{5} + \left(85 a^{2} + 22 a + 26\right)\cdot 97^{6} + \left(25 a^{2} + 20 a + 58\right)\cdot 97^{7} + \left(26 a^{2} + 32 a + 60\right)\cdot 97^{8} + \left(89 a^{2} + 35 a + 50\right)\cdot 97^{9} + \left(70 a^{2} + 37 a + 37\right)\cdot 97^{10} + \left(88 a^{2} + 53 a + 47\right)\cdot 97^{11} + \left(17 a^{2} + 27 a + 10\right)\cdot 97^{12} + \left(82 a^{2} + 11 a + 8\right)\cdot 97^{13} + \left(34 a^{2} + 26 a + 15\right)\cdot 97^{14} + \left(12 a^{2} + 51 a + 74\right)\cdot 97^{15} + \left(82 a^{2} + 77 a + 7\right)\cdot 97^{16} + \left(37 a^{2} + 29 a + 33\right)\cdot 97^{17} +O\left(97^{ 18 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 74 + 56\cdot 97 + 59\cdot 97^{2} + 57\cdot 97^{3} + 34\cdot 97^{4} + 2\cdot 97^{5} + 3\cdot 97^{6} + 49\cdot 97^{7} + 10\cdot 97^{8} + 23\cdot 97^{9} + 51\cdot 97^{10} + 58\cdot 97^{11} + 32\cdot 97^{12} + 81\cdot 97^{13} + 75\cdot 97^{14} + 6\cdot 97^{15} + 7\cdot 97^{16} + 96\cdot 97^{17} +O\left(97^{ 18 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a^{2} + 44 a + 47 + \left(49 a^{2} + 2 a + 4\right)\cdot 97 + \left(91 a^{2} + 6 a + 64\right)\cdot 97^{2} + \left(19 a^{2} + 30 a + 22\right)\cdot 97^{3} + \left(54 a^{2} + 36 a + 34\right)\cdot 97^{4} + \left(22 a^{2} + 34 a + 38\right)\cdot 97^{5} + \left(11 a^{2} + 11 a + 67\right)\cdot 97^{6} + \left(52 a^{2} + 14 a + 21\right)\cdot 97^{7} + \left(17 a^{2} + 36 a + 8\right)\cdot 97^{8} + \left(9 a^{2} + 10 a + 55\right)\cdot 97^{9} + \left(5 a^{2} + 53 a + 30\right)\cdot 97^{10} + \left(44 a^{2} + 63 a + 70\right)\cdot 97^{11} + \left(6 a^{2} + 58 a + 38\right)\cdot 97^{12} + \left(65 a^{2} + 96 a + 2\right)\cdot 97^{13} + \left(49 a^{2} + 68 a + 7\right)\cdot 97^{14} + \left(91 a^{2} + 11 a + 64\right)\cdot 97^{15} + \left(82 a^{2} + 43 a + 12\right)\cdot 97^{16} + \left(51 a^{2} + 79 a + 20\right)\cdot 97^{17} +O\left(97^{ 18 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 a^{2} + 55 a + 57 + \left(61 a^{2} + 11 a + 78\right)\cdot 97 + \left(94 a^{2} + 76 a + 82\right)\cdot 97^{2} + \left(35 a^{2} + 37 a + 21\right)\cdot 97^{3} + \left(89 a^{2} + 9 a + 51\right)\cdot 97^{4} + \left(44 a^{2} + 36 a + 75\right)\cdot 97^{5} + \left(73 a^{2} + 58 a + 52\right)\cdot 97^{6} + \left(27 a^{2} + 68 a + 69\right)\cdot 97^{7} + \left(2 a^{2} + 94 a + 13\right)\cdot 97^{8} + \left(55 a^{2} + 11 a + 39\right)\cdot 97^{9} + \left(23 a^{2} + 91 a + 44\right)\cdot 97^{10} + \left(49 a^{2} + 68 a + 4\right)\cdot 97^{11} + \left(60 a^{2} + 29 a + 72\right)\cdot 97^{12} + \left(a^{2} + 2 a + 9\right)\cdot 97^{13} + \left(52 a^{2} + 79 a + 21\right)\cdot 97^{14} + \left(56 a^{2} + 79 a + 48\right)\cdot 97^{15} + \left(48 a^{2} + 40 a\right)\cdot 97^{16} + \left(75 a^{2} + 96 a + 65\right)\cdot 97^{17} +O\left(97^{ 18 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 + 83\cdot 97 + 80\cdot 97^{2} + 16\cdot 97^{3} + 59\cdot 97^{4} + 39\cdot 97^{5} + 44\cdot 97^{6} + 77\cdot 97^{7} + 28\cdot 97^{8} + 95\cdot 97^{9} + 54\cdot 97^{10} + 51\cdot 97^{11} + 82\cdot 97^{12} + 66\cdot 97^{13} + 80\cdot 97^{14} + 84\cdot 97^{15} + 4\cdot 97^{16} + 52\cdot 97^{17} +O\left(97^{ 18 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 a^{2} + 85 a + 93 + \left(86 a^{2} + 10 a + 34\right)\cdot 97 + \left(15 a^{2} + 23 a + 95\right)\cdot 97^{2} + \left(96 a^{2} + 91 a + 91\right)\cdot 97^{3} + \left(55 a^{2} + 14 a + 44\right)\cdot 97^{4} + \left(4 a^{2} + 39 a + 27\right)\cdot 97^{5} + \left(34 a^{2} + 83 a + 10\right)\cdot 97^{6} + \left(66 a^{2} + 44 a + 10\right)\cdot 97^{7} + \left(73 a^{2} + 72 a + 54\right)\cdot 97^{8} + \left(85 a^{2} + 34 a + 29\right)\cdot 97^{9} + \left(31 a^{2} + 72 a + 94\right)\cdot 97^{10} + \left(6 a^{2} + 36 a + 37\right)\cdot 97^{11} + \left(93 a^{2} + 49 a + 73\right)\cdot 97^{12} + \left(82 a^{2} + 85 a + 12\right)\cdot 97^{13} + \left(48 a^{2} + 86 a + 2\right)\cdot 97^{14} + \left(73 a^{2} + 12 a + 53\right)\cdot 97^{15} + \left(82 a^{2} + 73 a + 11\right)\cdot 97^{16} + \left(6 a^{2} + 60 a + 41\right)\cdot 97^{17} +O\left(97^{ 18 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 90 a^{2} + 82 a + 55 + \left(50 a^{2} + 92 a + 14\right)\cdot 97 + \left(7 a^{2} + 32 a + 45\right)\cdot 97^{2} + \left(62 a^{2} + 3 a + 81\right)\cdot 97^{3} + \left(23 a^{2} + 39 a + 44\right)\cdot 97^{4} + \left(24 a^{2} + 18 a + 48\right)\cdot 97^{5} + \left(35 a^{2} + 16 a + 17\right)\cdot 97^{6} + \left(43 a^{2} + 8 a + 66\right)\cdot 97^{7} + \left(68 a^{2} + 67 a + 22\right)\cdot 97^{8} + \left(49 a^{2} + 49 a + 7\right)\cdot 97^{9} + \left(2 a^{2} + 65 a + 15\right)\cdot 97^{10} + \left(56 a^{2} + 71 a + 45\right)\cdot 97^{11} + \left(18 a^{2} + 39 a + 14\right)\cdot 97^{12} + \left(13 a^{2} + 83 a + 79\right)\cdot 97^{13} + \left(10 a^{2} + 88 a + 60\right)\cdot 97^{14} + \left(28 a^{2} + 62 a + 71\right)\cdot 97^{15} + \left(63 a^{2} + 75 a + 88\right)\cdot 97^{16} + \left(80 a^{2} + 67 a + 95\right)\cdot 97^{17} +O\left(97^{ 18 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 52 + 53\cdot 97 + 53\cdot 97^{2} + 22\cdot 97^{3} + 3\cdot 97^{4} + 55\cdot 97^{5} + 49\cdot 97^{6} + 67\cdot 97^{7} + 57\cdot 97^{8} + 75\cdot 97^{9} + 87\cdot 97^{10} + 83\cdot 97^{11} + 78\cdot 97^{12} + 45\cdot 97^{13} + 37\cdot 97^{14} + 5\cdot 97^{15} + 85\cdot 97^{16} + 45\cdot 97^{17} +O\left(97^{ 18 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 9 a^{2} + 65 a + 54 + \left(58 a^{2} + 83 a + 57\right)\cdot 97 + \left(86 a^{2} + 67 a + 34\right)\cdot 97^{2} + \left(77 a^{2} + 72 a + 79\right)\cdot 97^{3} + \left(83 a^{2} + 45 a + 17\right)\cdot 97^{4} + \left(69 a^{2} + 23 a + 31\right)\cdot 97^{5} + \left(51 a^{2} + 2 a + 19\right)\cdot 97^{6} + \left(75 a^{2} + 38 a + 65\right)\cdot 97^{7} + \left(5 a^{2} + 85 a + 34\right)\cdot 97^{8} + \left(2 a^{2} + 51 a + 12\right)\cdot 97^{9} + \left(60 a^{2} + 68 a + 69\right)\cdot 97^{10} + \left(46 a^{2} + 93 a + 85\right)\cdot 97^{11} + \left(94 a^{2} + 85 a + 81\right)\cdot 97^{12} + \left(45 a^{2} + 11 a + 81\right)\cdot 97^{13} + \left(95 a^{2} + 38 a + 87\right)\cdot 97^{14} + \left(28 a^{2} + 72 a + 76\right)\cdot 97^{15} + \left(28 a^{2} + 77 a + 72\right)\cdot 97^{16} + \left(38 a^{2} + 53 a + 35\right)\cdot 97^{17} +O\left(97^{ 18 }\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.
