Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 14.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{3} + 9 x + 92 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 a^{2} + 9 a + 3 + \left(40 a^{2} + 72 a + 49\right)\cdot 97 + \left(64 a^{2} + 81 a + 95\right)\cdot 97^{2} + \left(34 a^{2} + 72 a + 13\right)\cdot 97^{3} + \left(38 a^{2} + 93 a + 36\right)\cdot 97^{4} + \left(18 a^{2} + 82 a + 13\right)\cdot 97^{5} + \left(67 a^{2} + 42 a + 15\right)\cdot 97^{6} + \left(63 a^{2} + 62 a + 91\right)\cdot 97^{7} + \left(95 a^{2} + 36 a + 88\right)\cdot 97^{8} + \left(47 a^{2} + 14 a + 93\right)\cdot 97^{9} + \left(43 a^{2} + 25 a + 66\right)\cdot 97^{10} + \left(12 a^{2} + 43 a + 74\right)\cdot 97^{11} + \left(27 a^{2} + 48 a + 65\right)\cdot 97^{12} + \left(17 a^{2} + 53 a + 6\right)\cdot 97^{13} +O\left(97^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 74 a^{2} + 20 a + 56 + \left(90 a^{2} + 22 a + 59\right)\cdot 97 + \left(23 a^{2} + 19 a + 46\right)\cdot 97^{2} + \left(48 a^{2} + 66 a + 95\right)\cdot 97^{3} + \left(41 a^{2} + 94 a + 54\right)\cdot 97^{4} + \left(93 a^{2} + 80 a + 75\right)\cdot 97^{5} + \left(69 a^{2} + 46 a + 31\right)\cdot 97^{6} + \left(19 a^{2} + 31 a + 21\right)\cdot 97^{7} + \left(72 a^{2} + 61 a + 45\right)\cdot 97^{8} + \left(30 a^{2} + 79 a + 87\right)\cdot 97^{9} + \left(47 a^{2} + 73 a + 89\right)\cdot 97^{10} + \left(47 a^{2} + 35 a + 90\right)\cdot 97^{11} + \left(6 a^{2} + 22 a + 38\right)\cdot 97^{12} + \left(24 a^{2} + 69 a + 47\right)\cdot 97^{13} +O\left(97^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 97 + 48\cdot 97^{2} + 81\cdot 97^{3} + 41\cdot 97^{4} + 13\cdot 97^{5} + 31\cdot 97^{6} + 2\cdot 97^{7} + 27\cdot 97^{8} + 61\cdot 97^{9} + 79\cdot 97^{10} + 19\cdot 97^{11} + 97^{12} + 53\cdot 97^{13} +O\left(97^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 66 a^{2} + 24 a + 8 + \left(35 a^{2} + 54 a + 20\right)\cdot 97 + \left(16 a^{2} + 25 a + 1\right)\cdot 97^{2} + \left(85 a^{2} + 89 a + 26\right)\cdot 97^{3} + \left(79 a^{2} + a + 91\right)\cdot 97^{4} + \left(96 a^{2} + 36 a + 95\right)\cdot 97^{5} + \left(84 a^{2} + 69 a + 24\right)\cdot 97^{6} + \left(73 a^{2} + 8 a + 55\right)\cdot 97^{7} + \left(68 a^{2} + 56 a + 24\right)\cdot 97^{8} + \left(83 a^{2} + 16 a + 17\right)\cdot 97^{9} + \left(83 a^{2} + 33 a + 18\right)\cdot 97^{10} + \left(76 a^{2} + 43 a + 73\right)\cdot 97^{11} + \left(15 a^{2} + 65 a + 94\right)\cdot 97^{12} + \left(69 a^{2} + 75 a + 26\right)\cdot 97^{13} +O\left(97^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 a^{2} + 21 a + 20 + \left(12 a^{2} + 11 a + 76\right)\cdot 97 + \left(14 a^{2} + 69 a + 84\right)\cdot 97^{2} + \left(32 a^{2} + 9 a + 95\right)\cdot 97^{3} + \left(11 a^{2} + 80 a + 67\right)\cdot 97^{4} + \left(28 a^{2} + 7 a + 71\right)\cdot 97^{5} + \left(96 a^{2} + 94 a + 92\right)\cdot 97^{6} + \left(63 a^{2} + 64 a + 92\right)\cdot 97^{7} + \left(95 a^{2} + 12 a + 88\right)\cdot 97^{8} + \left(25 a^{2} + 69 a + 58\right)\cdot 97^{9} + \left(49 a^{2} + 93 a + 4\right)\cdot 97^{10} + \left(53 a^{2} + 29 a + 30\right)\cdot 97^{11} + \left(80 a^{2} + 89 a + 95\right)\cdot 97^{12} + \left(19 a^{2} + 3 a + 21\right)\cdot 97^{13} +O\left(97^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 + 64\cdot 97 + 53\cdot 97^{2} + 71\cdot 97^{3} + 33\cdot 97^{4} + 76\cdot 97^{5} + 20\cdot 97^{6} + 38\cdot 97^{7} + 18\cdot 97^{8} + 8\cdot 97^{9} + 29\cdot 97^{10} + 17\cdot 97^{11} + 84\cdot 97^{12} + 80\cdot 97^{13} +O\left(97^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 79 a^{2} + 64 a + 86 + \left(20 a^{2} + 67 a + 27\right)\cdot 97 + \left(16 a^{2} + 86 a\right)\cdot 97^{2} + \left(74 a^{2} + 31 a + 57\right)\cdot 97^{3} + \left(75 a^{2} + a + 66\right)\cdot 97^{4} + \left(78 a^{2} + 75 a + 84\right)\cdot 97^{5} + \left(41 a^{2} + 81 a + 56\right)\cdot 97^{6} + \left(56 a^{2} + 25 a + 47\right)\cdot 97^{7} + \left(29 a^{2} + 4 a + 80\right)\cdot 97^{8} + \left(62 a^{2} + 66 a + 82\right)\cdot 97^{9} + \left(66 a^{2} + 38 a + 11\right)\cdot 97^{10} + \left(7 a^{2} + 10 a + 46\right)\cdot 97^{11} + \left(54 a^{2} + 80 a + 33\right)\cdot 97^{12} + \left(10 a^{2} + 64 a + 63\right)\cdot 97^{13} +O\left(97^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 52 a^{2} + 56 a + 21 + \left(90 a^{2} + 63 a + 58\right)\cdot 97 + \left(58 a^{2} + 8 a + 62\right)\cdot 97^{2} + \left(16 a^{2} + 21 a + 2\right)\cdot 97^{3} + \left(44 a^{2} + 19 a + 71\right)\cdot 97^{4} + \left(72 a^{2} + 8 a + 46\right)\cdot 97^{5} + \left(27 a^{2} + 53 a + 69\right)\cdot 97^{6} + \left(13 a^{2} + 79\right)\cdot 97^{7} + \left(26 a^{2} + 23 a + 59\right)\cdot 97^{8} + \left(40 a^{2} + 45 a + 47\right)\cdot 97^{9} + \left(26 a + 2\right)\cdot 97^{10} + \left(93 a^{2} + 31 a + 73\right)\cdot 97^{11} + \left(9 a^{2} + 82 a + 59\right)\cdot 97^{12} + \left(53 a^{2} + 23 a + 27\right)\cdot 97^{13} +O\left(97^{ 14 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 37 + 31\cdot 97 + 92\cdot 97^{2} + 40\cdot 97^{3} + 21\cdot 97^{4} + 7\cdot 97^{5} + 45\cdot 97^{6} + 56\cdot 97^{7} + 51\cdot 97^{8} + 27\cdot 97^{9} + 85\cdot 97^{10} + 59\cdot 97^{11} + 11\cdot 97^{12} + 60\cdot 97^{13} +O\left(97^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2)(4,6,5)(7,9,8)$ |
| $(1,7,4)(2,8,5)(3,9,6)$ |
| $(2,6)(3,5)(4,7)(8,9)$ |
| $(2,5,8)(3,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,7,4)(2,8,5)(3,9,6)$ | $-3$ |
| $3$ | $3$ | $(1,7,4)(2,5,8)$ | $0$ |
| $3$ | $3$ | $(1,4,7)(2,8,5)$ | $0$ |
| $6$ | $3$ | $(1,3,2)(4,6,5)(7,9,8)$ | $0$ |
| $6$ | $3$ | $(1,2,3)(4,5,6)(7,8,9)$ | $0$ |
| $6$ | $3$ | $(1,6,2)(3,8,7)(4,9,5)$ | $0$ |
| $9$ | $6$ | $(1,5,7,8,4,2)(3,6)$ | $0$ |
| $9$ | $6$ | $(1,2,4,8,7,5)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
