Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{3} + 9 x + 92 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 a^{2} + 77 a + 44 + \left(8 a^{2} + 23 a + 63\right)\cdot 97 + \left(20 a^{2} + 56 a + 11\right)\cdot 97^{2} + \left(15 a^{2} + 38 a + 37\right)\cdot 97^{3} + \left(10 a^{2} + 96 a + 77\right)\cdot 97^{4} + \left(6 a^{2} + 6 a + 62\right)\cdot 97^{5} + \left(3 a^{2} + 84 a\right)\cdot 97^{6} + \left(62 a^{2} + 37 a + 47\right)\cdot 97^{7} + \left(11 a^{2} + 54 a + 35\right)\cdot 97^{8} + \left(20 a^{2} + 16 a + 4\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 93 + 11\cdot 97 + 45\cdot 97^{2} + 62\cdot 97^{3} + 17\cdot 97^{4} + 13\cdot 97^{5} + 33\cdot 97^{6} + 34\cdot 97^{7} + 91\cdot 97^{8} + 86\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 81 a^{2} + 18 a + 72 + \left(90 a^{2} + 38 a + 57\right)\cdot 97 + \left(6 a^{2} + 13 a + 60\right)\cdot 97^{2} + \left(60 a^{2} + 60 a + 89\right)\cdot 97^{3} + \left(82 a^{2} + 76 a + 27\right)\cdot 97^{4} + \left(86 a^{2} + 46 a + 61\right)\cdot 97^{5} + \left(10 a^{2} + 41 a + 91\right)\cdot 97^{6} + \left(27 a^{2} + 13 a + 33\right)\cdot 97^{7} + \left(11 a^{2} + 45 a + 34\right)\cdot 97^{8} + \left(35 a^{2} + 55 a + 80\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a^{2} + 32 a + 3 + \left(61 a^{2} + 65 a + 74\right)\cdot 97 + \left(41 a^{2} + 12 a + 74\right)\cdot 97^{2} + \left(61 a^{2} + 23 a\right)\cdot 97^{3} + \left(44 a^{2} + 31 a + 91\right)\cdot 97^{4} + \left(63 a^{2} + 2 a + 17\right)\cdot 97^{5} + \left(38 a^{2} + 94 a + 64\right)\cdot 97^{6} + \left(95 a^{2} + 9 a + 55\right)\cdot 97^{7} + \left(29 a^{2} + 79 a + 49\right)\cdot 97^{8} + \left(40 a^{2} + 10 a + 14\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 92 + 23\cdot 97 + 67\cdot 97^{2} + 18\cdot 97^{3} + 91\cdot 97^{4} + 39\cdot 97^{5} + 71\cdot 97^{6} + 77\cdot 97^{8} +O\left(97^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 a^{2} + 94 a + 80 + \left(15 a^{2} + 61 a + 8\right)\cdot 97 + \left(30 a^{2} + 44 a + 72\right)\cdot 97^{2} + \left(28 a^{2} + 85 a + 18\right)\cdot 97^{3} + \left(55 a^{2} + 24 a + 57\right)\cdot 97^{4} + \left(89 a^{2} + 77 a + 78\right)\cdot 97^{5} + \left(42 a^{2} + 21 a + 45\right)\cdot 97^{6} + \left(46 a^{2} + 86 a + 50\right)\cdot 97^{7} + \left(64 a^{2} + 56 a + 61\right)\cdot 97^{8} + \left(36 a^{2} + 37 a + 6\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 92 a^{2} + 47 a + 41 + \left(41 a^{2} + 90 a + 55\right)\cdot 97 + \left(48 a^{2} + 70 a + 18\right)\cdot 97^{2} + \left(72 a^{2} + 13 a + 67\right)\cdot 97^{3} + \left(66 a^{2} + 86 a + 29\right)\cdot 97^{4} + \left(43 a^{2} + 47 a + 93\right)\cdot 97^{5} + \left(47 a^{2} + 58 a + 19\right)\cdot 97^{6} + \left(71 a^{2} + 73 a + 9\right)\cdot 97^{7} + \left(55 a^{2} + 69 a + 10\right)\cdot 97^{8} + \left(21 a^{2} + 30 a + 96\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 63 + 30\cdot 97 + 60\cdot 97^{2} + 20\cdot 97^{3} + 81\cdot 97^{4} + 83\cdot 97^{5} + 63\cdot 97^{6} + 65\cdot 97^{7} + 34\cdot 97^{8} + 70\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 90 a^{2} + 23 a + 96 + \left(72 a^{2} + 11 a + 61\right)\cdot 97 + \left(46 a^{2} + 93 a + 74\right)\cdot 97^{2} + \left(53 a^{2} + 69 a + 72\right)\cdot 97^{3} + \left(31 a^{2} + 72 a + 11\right)\cdot 97^{4} + \left(a^{2} + 12 a + 34\right)\cdot 97^{5} + \left(51 a^{2} + 88 a + 94\right)\cdot 97^{6} + \left(85 a^{2} + 69 a + 90\right)\cdot 97^{7} + \left(20 a^{2} + 82 a + 90\right)\cdot 97^{8} + \left(40 a^{2} + 42 a + 27\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,4,2)(3,8,6)(5,9,7)$ |
| $(2,5,8)(3,7,4)$ |
| $(2,3)(4,8)(5,7)(6,9)$ |
| $(1,9,6)(2,5,8)(3,4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,5)(2,9)(3,4)(6,8)$ | $0$ |
| $2$ | $3$ | $(1,9,6)(2,5,8)(3,4,7)$ | $-3$ |
| $3$ | $3$ | $(1,9,6)(2,8,5)$ | $0$ |
| $3$ | $3$ | $(1,6,9)(2,5,8)$ | $0$ |
| $6$ | $3$ | $(1,4,2)(3,8,6)(5,9,7)$ | $0$ |
| $6$ | $3$ | $(1,2,4)(3,6,8)(5,7,9)$ | $0$ |
| $6$ | $3$ | $(1,3,2)(4,5,9)(6,7,8)$ | $0$ |
| $9$ | $6$ | $(1,8,9,5,6,2)(3,4)$ | $0$ |
| $9$ | $6$ | $(1,2,6,5,9,8)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
