Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{3} + 9 x + 92 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 83 a^{2} + 21 a + 13 + \left(71 a^{2} + 65 a + 43\right)\cdot 97 + \left(18 a^{2} + 15 a + 15\right)\cdot 97^{2} + \left(32 a^{2} + 49 a + 96\right)\cdot 97^{3} + \left(73 a^{2} + 81 a + 51\right)\cdot 97^{4} + \left(25 a^{2} + 81 a + 57\right)\cdot 97^{5} + \left(35 a^{2} + 78 a + 17\right)\cdot 97^{6} + \left(35 a^{2} + 8 a + 18\right)\cdot 97^{7} + \left(93 a^{2} + 91 a + 75\right)\cdot 97^{8} + \left(29 a^{2} + 41 a + 82\right)\cdot 97^{9} + \left(89 a^{2} + 57 a + 50\right)\cdot 97^{10} +O\left(97^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 63 a^{2} + 33 a + 87 + \left(84 a^{2} + 27 a + 22\right)\cdot 97 + \left(21 a^{2} + 60 a + 34\right)\cdot 97^{2} + \left(86 a^{2} + 70 a + 32\right)\cdot 97^{3} + \left(18 a^{2} + 86 a + 16\right)\cdot 97^{4} + \left(79 a^{2} + a + 87\right)\cdot 97^{5} + \left(47 a^{2} + a + 92\right)\cdot 97^{6} + \left(79 a^{2} + 44 a + 88\right)\cdot 97^{7} + \left(18 a^{2} + 40 a + 15\right)\cdot 97^{8} + \left(35 a^{2} + 86 a + 17\right)\cdot 97^{9} + \left(67 a^{2} + 74 a + 16\right)\cdot 97^{10} +O\left(97^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 57\cdot 97 + 16\cdot 97^{2} + 96\cdot 97^{3} + 32\cdot 97^{4} + 24\cdot 97^{5} + 20\cdot 97^{6} + 58\cdot 97^{7} + 56\cdot 97^{8} + 71\cdot 97^{9} + 84\cdot 97^{10} +O\left(97^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 6\cdot 97 + 38\cdot 97^{2} + 31\cdot 97^{3} + 23\cdot 97^{4} + 90\cdot 97^{5} + 54\cdot 97^{6} + 19\cdot 97^{7} + 22\cdot 97^{8} + 78\cdot 97^{9} + 58\cdot 97^{10} +O\left(97^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a^{2} + 79 a + 71 + \left(37 a^{2} + 89 a + 29\right)\cdot 97 + \left(50 a^{2} + 5 a + 11\right)\cdot 97^{2} + \left(22 a^{2} + 84 a + 38\right)\cdot 97^{3} + \left(26 a^{2} + 39 a + 60\right)\cdot 97^{4} + \left(21 a^{2} + 88 a + 30\right)\cdot 97^{5} + \left(84 a^{2} + 8 a + 20\right)\cdot 97^{6} + \left(94 a^{2} + 26 a + 84\right)\cdot 97^{7} + \left(81 a^{2} + 25 a + 6\right)\cdot 97^{8} + \left(59 a^{2} + 87 a + 68\right)\cdot 97^{9} + \left(49 a^{2} + 10 a + 6\right)\cdot 97^{10} +O\left(97^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a^{2} + 17 a + 16 + \left(74 a^{2} + 94 a + 58\right)\cdot 97 + \left(18 a^{2} + 41 a + 15\right)\cdot 97^{2} + \left(55 a^{2} + 26 a + 40\right)\cdot 97^{3} + \left(60 a^{2} + 29 a + 72\right)\cdot 97^{4} + \left(25 a^{2} + 11 a + 56\right)\cdot 97^{5} + \left(75 a^{2} + 29 a + 63\right)\cdot 97^{6} + \left(44 a^{2} + 69 a + 74\right)\cdot 97^{7} + \left(25 a^{2} + 66 a + 55\right)\cdot 97^{8} + \left(26 a^{2} + 74 a + 60\right)\cdot 97^{9} + \left(71 a^{2} + 45 a + 39\right)\cdot 97^{10} +O\left(97^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 + 33\cdot 97 + 42\cdot 97^{2} + 66\cdot 97^{3} + 40\cdot 97^{4} + 79\cdot 97^{5} + 21\cdot 97^{6} + 19\cdot 97^{7} + 18\cdot 97^{8} + 44\cdot 97^{9} + 50\cdot 97^{10} +O\left(97^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 6 a^{2} + 82 a + 36 + \left(72 a^{2} + 76 a + 44\right)\cdot 97 + \left(24 a^{2} + 30 a + 51\right)\cdot 97^{2} + \left(85 a^{2} + 39 a + 26\right)\cdot 97^{3} + \left(51 a^{2} + 67 a + 20\right)\cdot 97^{4} + \left(93 a^{2} + 6 a + 76\right)\cdot 97^{5} + \left(61 a^{2} + 87 a + 80\right)\cdot 97^{6} + \left(19 a^{2} + 26 a + 20\right)\cdot 97^{7} + \left(93 a^{2} + 31 a + 74\right)\cdot 97^{8} + \left(a^{2} + 20 a + 11\right)\cdot 97^{9} + \left(77 a^{2} + 11 a + 74\right)\cdot 97^{10} +O\left(97^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 76 a^{2} + 59 a + 68 + \left(47 a^{2} + 34 a + 92\right)\cdot 97 + \left(59 a^{2} + 39 a + 65\right)\cdot 97^{2} + \left(9 a^{2} + 21 a + 57\right)\cdot 97^{3} + \left(60 a^{2} + 83 a + 69\right)\cdot 97^{4} + \left(45 a^{2} + 3 a + 79\right)\cdot 97^{5} + \left(83 a^{2} + 86 a + 15\right)\cdot 97^{6} + \left(16 a^{2} + 18 a + 4\right)\cdot 97^{7} + \left(75 a^{2} + 36 a + 63\right)\cdot 97^{8} + \left(40 a^{2} + 77 a + 50\right)\cdot 97^{9} + \left(33 a^{2} + 90 a + 6\right)\cdot 97^{10} +O\left(97^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,5,8)(3,4,7)$ |
| $(2,5)(4,7)(6,9)$ |
| $(1,9,6)(2,5,8)(3,7,4)$ |
| $(1,3,2)(4,8,6)(5,9,7)$ |
| $(2,3)(4,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,3)(4,8)(5,7)$ | $-2$ |
| $9$ | $2$ | $(2,4)(3,8)(5,7)(6,9)$ | $0$ |
| $9$ | $2$ | $(2,5)(4,7)(6,9)$ | $0$ |
| $2$ | $3$ | $(1,9,6)(2,5,8)(3,7,4)$ | $-3$ |
| $6$ | $3$ | $(1,4,2)(3,5,9)(6,7,8)$ | $0$ |
| $6$ | $3$ | $(2,5,8)(3,4,7)$ | $0$ |
| $12$ | $3$ | $(1,3,2)(4,8,6)(5,9,7)$ | $0$ |
| $18$ | $6$ | $(1,9,6)(2,7,8,3,5,4)$ | $1$ |
| $18$ | $6$ | $(2,3,5,4,8,7)(6,9)$ | $0$ |
| $18$ | $6$ | $(1,3,2,9,4,5)(6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
