Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{3} + 9 x + 92 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 34 a^{2} + 77 a + 30 + \left(32 a^{2} + 89 a + 68\right)\cdot 97 + \left(59 a^{2} + 27 a + 19\right)\cdot 97^{2} + \left(22 a^{2} + 57 a + 12\right)\cdot 97^{3} + \left(28 a^{2} + 40 a + 29\right)\cdot 97^{4} + \left(60 a^{2} + 60 a + 28\right)\cdot 97^{5} + \left(6 a^{2} + 19 a + 67\right)\cdot 97^{6} + \left(78 a^{2} + 35 a + 94\right)\cdot 97^{7} + \left(92 a^{2} + 31 a + 11\right)\cdot 97^{8} + \left(50 a^{2} + 31 a + 56\right)\cdot 97^{9} + \left(96 a^{2} + 30 a + 89\right)\cdot 97^{10} + \left(61 a^{2} + 48 a + 35\right)\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a^{2} + 43 a + 61 + \left(51 a^{2} + 40 a + 52\right)\cdot 97 + \left(72 a^{2} + 2 a + 17\right)\cdot 97^{2} + \left(78 a^{2} + a + 43\right)\cdot 97^{3} + \left(32 a^{2} + 82 a + 96\right)\cdot 97^{4} + \left(27 a^{2} + 55 a + 91\right)\cdot 97^{5} + \left(46 a^{2} + 17 a + 39\right)\cdot 97^{6} + \left(80 a^{2} + 28 a + 15\right)\cdot 97^{7} + \left(34 a^{2} + 9 a + 19\right)\cdot 97^{8} + \left(71 a^{2} + 68 a + 9\right)\cdot 97^{9} + \left(a^{2} + 78 a + 10\right)\cdot 97^{10} + \left(62 a^{2} + 41 a + 41\right)\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 + 75\cdot 97 + 92\cdot 97^{2} + 5\cdot 97^{3} + 33\cdot 97^{4} + 44\cdot 97^{5} + 42\cdot 97^{6} + 28\cdot 97^{7} + 87\cdot 97^{8} + 25\cdot 97^{9} + 65\cdot 97^{10} + 20\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 54\cdot 97 + 25\cdot 97^{2} + 6\cdot 97^{3} + 17\cdot 97^{4} + 47\cdot 97^{5} + 67\cdot 97^{6} + 85\cdot 97^{7} + 87\cdot 97^{9} + 34\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a^{2} + 88 a + 61 + \left(88 a^{2} + 23 a + 80\right)\cdot 97 + \left(87 a^{2} + 39 a + 12\right)\cdot 97^{2} + \left(80 a^{2} + 38 a + 56\right)\cdot 97^{3} + \left(82 a^{2} + 96 a + 8\right)\cdot 97^{4} + \left(47 a^{2} + 34 a + 21\right)\cdot 97^{5} + \left(91 a^{2} + 16 a + 20\right)\cdot 97^{6} + \left(58 a^{2} + 44 a + 80\right)\cdot 97^{7} + \left(66 a^{2} + 32 a + 15\right)\cdot 97^{8} + \left(22 a^{2} + 60 a + 8\right)\cdot 97^{9} + \left(82 a^{2} + 87 a + 8\right)\cdot 97^{10} + \left(80 a^{2} + 26 a + 57\right)\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 95 a^{2} + 63 a + 43 + \left(54 a^{2} + 32 a + 76\right)\cdot 97 + \left(33 a^{2} + 55 a + 74\right)\cdot 97^{2} + \left(34 a^{2} + 57 a + 67\right)\cdot 97^{3} + \left(78 a^{2} + 15 a + 78\right)\cdot 97^{4} + \left(21 a^{2} + 6 a + 58\right)\cdot 97^{5} + \left(56 a^{2} + 63 a + 2\right)\cdot 97^{6} + \left(54 a^{2} + 24 a + 54\right)\cdot 97^{7} + \left(92 a^{2} + 55 a + 74\right)\cdot 97^{8} + \left(2 a^{2} + 65 a + 83\right)\cdot 97^{9} + \left(13 a^{2} + 27 a + 77\right)\cdot 97^{10} + \left(51 a^{2} + 28 a + 72\right)\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 68 + 37\cdot 97 + 9\cdot 97^{2} + 94\cdot 97^{3} + 89\cdot 97^{4} + 57\cdot 97^{5} + 35\cdot 97^{6} + 81\cdot 97^{7} + 78\cdot 97^{8} + 49\cdot 97^{9} + 45\cdot 97^{10} + 6\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 a^{2} + 20 a + 28 + \left(23 a^{2} + 55 a + 16\right)\cdot 97 + \left(94 a^{2} + 65 a + 35\right)\cdot 97^{2} + \left(48 a^{2} + 25 a + 73\right)\cdot 97^{3} + \left(88 a^{2} + 66 a + 2\right)\cdot 97^{4} + \left(48 a^{2} + 59 a + 57\right)\cdot 97^{5} + \left(25 a^{2} + 59 a + 83\right)\cdot 97^{6} + \left(42 a^{2} + 31 a + 73\right)\cdot 97^{7} + \left(70 a^{2} + 72 a + 71\right)\cdot 97^{8} + \left(9 a^{2} + 59 a + 2\right)\cdot 97^{9} + \left(77 a^{2} + 73 a + 70\right)\cdot 97^{10} + \left(32 a^{2} + 86 a + 54\right)\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 94 a^{2} + 2 + \left(40 a^{2} + 49 a + 23\right)\cdot 97 + \left(40 a^{2} + 3 a + 3\right)\cdot 97^{2} + \left(25 a^{2} + 14 a + 29\right)\cdot 97^{3} + \left(77 a^{2} + 87 a + 32\right)\cdot 97^{4} + \left(84 a^{2} + 73 a + 78\right)\cdot 97^{5} + \left(64 a^{2} + 17 a + 28\right)\cdot 97^{6} + \left(73 a^{2} + 30 a + 68\right)\cdot 97^{7} + \left(30 a^{2} + 90 a + 27\right)\cdot 97^{8} + \left(36 a^{2} + 5 a + 65\right)\cdot 97^{9} + \left(20 a^{2} + 90 a + 20\right)\cdot 97^{10} + \left(2 a^{2} + 58 a + 65\right)\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,9,8)(2,6,5)(3,4,7)$ |
| $(1,8)(2,3)(4,5)(6,7)$ |
| $(2,6,5)(3,7,4)$ |
| $(1,3,5)(2,9,4)(6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,5)(2,8)(3,4)(6,9)$ | $0$ |
| $2$ | $3$ | $(1,9,8)(2,6,5)(3,4,7)$ | $-3$ |
| $3$ | $3$ | $(1,9,8)(2,5,6)$ | $0$ |
| $3$ | $3$ | $(1,8,9)(2,6,5)$ | $0$ |
| $6$ | $3$ | $(1,3,5)(2,9,4)(6,8,7)$ | $0$ |
| $6$ | $3$ | $(1,5,3)(2,4,9)(6,7,8)$ | $0$ |
| $6$ | $3$ | $(1,7,5)(2,9,3)(4,6,8)$ | $0$ |
| $9$ | $6$ | $(1,2,9,5,8,6)(3,4)$ | $0$ |
| $9$ | $6$ | $(1,6,8,5,9,2)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
