Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{3} + 9 x + 92 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 a^{2} + 15 a + 42 + \left(85 a^{2} + 46 a + 25\right)\cdot 97 + \left(11 a^{2} + 87 a + 71\right)\cdot 97^{2} + \left(66 a^{2} + 61 a + 8\right)\cdot 97^{3} + \left(69 a^{2} + 40 a + 30\right)\cdot 97^{4} + \left(84 a^{2} + 69 a + 23\right)\cdot 97^{5} + \left(5 a^{2} + 34 a + 35\right)\cdot 97^{6} + \left(21 a^{2} + 41 a + 29\right)\cdot 97^{7} + \left(94 a^{2} + 25 a + 80\right)\cdot 97^{8} + \left(81 a^{2} + 71 a + 6\right)\cdot 97^{9} + \left(34 a^{2} + 32 a + 15\right)\cdot 97^{10} + \left(59 a^{2} + 55 a + 65\right)\cdot 97^{11} + \left(19 a^{2} + 4 a + 20\right)\cdot 97^{12} +O\left(97^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 73 + 51\cdot 97 + 8\cdot 97^{2} + 43\cdot 97^{3} + 71\cdot 97^{4} + 33\cdot 97^{5} + 39\cdot 97^{6} + 11\cdot 97^{7} + 16\cdot 97^{8} + 53\cdot 97^{9} + 17\cdot 97^{10} + 20\cdot 97^{11} + 74\cdot 97^{12} +O\left(97^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 a^{2} + 67 a + 87 + \left(5 a^{2} + 60 a + 33\right)\cdot 97 + \left(67 a^{2} + 92 a + 14\right)\cdot 97^{2} + \left(29 a^{2} + 59 a + 81\right)\cdot 97^{3} + \left(90 a^{2} + 66 a + 56\right)\cdot 97^{4} + \left(24 a^{2} + 7 a + 52\right)\cdot 97^{5} + \left(95 a^{2} + 26 a + 86\right)\cdot 97^{6} + \left(78 a^{2} + 64 a + 85\right)\cdot 97^{7} + \left(64 a^{2} + 82 a\right)\cdot 97^{8} + \left(88 a^{2} + 46 a + 47\right)\cdot 97^{9} + \left(6 a^{2} + 15 a + 41\right)\cdot 97^{10} + \left(94 a^{2} + 24 a + 79\right)\cdot 97^{11} + \left(95 a^{2} + 23 a + 90\right)\cdot 97^{12} +O\left(97^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 a^{2} + 40 a + 15 + \left(22 a^{2} + 22 a + 38\right)\cdot 97 + \left(73 a^{2} + 77 a + 51\right)\cdot 97^{2} + \left(24 a^{2} + 74 a + 51\right)\cdot 97^{3} + \left(87 a^{2} + 70 a + 38\right)\cdot 97^{4} + \left(17 a^{2} + 4 a + 10\right)\cdot 97^{5} + \left(22 a^{2} + 19 a + 36\right)\cdot 97^{6} + \left(78 a^{2} + 81\right)\cdot 97^{7} + \left(59 a^{2} + 30 a + 67\right)\cdot 97^{8} + \left(34 a^{2} + 61 a + 13\right)\cdot 97^{9} + \left(36 a^{2} + 64 a + 24\right)\cdot 97^{10} + \left(59 a^{2} + 72 a + 65\right)\cdot 97^{11} + \left(74 a^{2} + 90 a + 59\right)\cdot 97^{12} +O\left(97^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 53\cdot 97 + 92\cdot 97^{2} + 96\cdot 97^{3} + 38\cdot 97^{4} + 8\cdot 97^{6} + 41\cdot 97^{7} + 47\cdot 97^{8} + 19\cdot 97^{9} + 34\cdot 97^{10} + 35\cdot 97^{11} + 18\cdot 97^{12} +O\left(97^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 a^{2} + 17 a + 38 + \left(83 a^{2} + 72 a + 17\right)\cdot 97 + \left(48 a^{2} + 23 a + 2\right)\cdot 97^{2} + \left(86 a + 3\right)\cdot 97^{3} + \left(71 a^{2} + 68 a + 38\right)\cdot 97^{4} + \left(14 a^{2} + 27 a + 88\right)\cdot 97^{5} + \left(17 a^{2} + 73 a + 5\right)\cdot 97^{6} + \left(44 a^{2} + 28 a + 71\right)\cdot 97^{7} + \left(14 a^{2} + 83 a + 86\right)\cdot 97^{8} + \left(60 a^{2} + 69 a + 69\right)\cdot 97^{9} + \left(56 a^{2} + 63 a + 48\right)\cdot 97^{10} + \left(42 a^{2} + 26 a + 61\right)\cdot 97^{11} + \left(94 a^{2} + 83 a + 81\right)\cdot 97^{12} +O\left(97^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 a^{2} + 42 a + 40 + \left(86 a^{2} + 28 a + 33\right)\cdot 97 + \left(11 a^{2} + 29 a + 71\right)\cdot 97^{2} + \left(6 a^{2} + 57 a + 36\right)\cdot 97^{3} + \left(37 a^{2} + 82 a + 28\right)\cdot 97^{4} + \left(91 a^{2} + 22 a + 63\right)\cdot 97^{5} + \left(68 a^{2} + 43 a + 25\right)\cdot 97^{6} + \left(94 a^{2} + 55 a + 83\right)\cdot 97^{7} + \left(39 a^{2} + 41 a + 45\right)\cdot 97^{8} + \left(77 a^{2} + 61 a + 76\right)\cdot 97^{9} + \left(25 a^{2} + 96 a + 57\right)\cdot 97^{10} + \left(75 a^{2} + 65 a + 63\right)\cdot 97^{11} + \left(2 a^{2} + a + 16\right)\cdot 97^{12} +O\left(97^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 88 + 88\cdot 97 + 92\cdot 97^{2} + 53\cdot 97^{3} + 83\cdot 97^{4} + 62\cdot 97^{5} + 49\cdot 97^{6} + 44\cdot 97^{7} + 33\cdot 97^{8} + 24\cdot 97^{9} + 45\cdot 97^{10} + 41\cdot 97^{11} + 4\cdot 97^{12} +O\left(97^{ 13 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 60 a^{2} + 13 a + 69 + \left(7 a^{2} + 61 a + 45\right)\cdot 97 + \left(78 a^{2} + 77 a + 80\right)\cdot 97^{2} + \left(66 a^{2} + 47 a + 12\right)\cdot 97^{3} + \left(32 a^{2} + 58 a + 2\right)\cdot 97^{4} + \left(57 a^{2} + 61 a + 53\right)\cdot 97^{5} + \left(81 a^{2} + 94 a + 4\right)\cdot 97^{6} + \left(70 a^{2} + 3 a + 37\right)\cdot 97^{7} + \left(17 a^{2} + 28 a + 9\right)\cdot 97^{8} + \left(45 a^{2} + 77 a + 77\right)\cdot 97^{9} + \left(33 a^{2} + 17 a + 6\right)\cdot 97^{10} + \left(57 a^{2} + 46 a + 53\right)\cdot 97^{11} + \left(3 a^{2} + 87 a + 21\right)\cdot 97^{12} +O\left(97^{ 13 }\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,2)(3,6)(4,8)(5,7)$ |
$0$ |
| $2$ |
$3$ |
$(1,4,7)(2,5,8)(3,6,9)$ |
$-3$ |
| $3$ |
$3$ |
$(1,7,4)(2,5,8)$ |
$0$ |
| $3$ |
$3$ |
$(1,4,7)(2,8,5)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,5)(2,7,9)(4,6,8)$ |
$0$ |
| $6$ |
$3$ |
$(1,5,3)(2,9,7)(4,8,6)$ |
$0$ |
| $6$ |
$3$ |
$(1,5,6)(2,3,7)(4,8,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,8,7,2,4,5)(3,6)$ |
$0$ |
| $9$ |
$6$ |
$(1,5,4,2,7,8)(3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
