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:
| $r_{ 1 }$ |
$=$ |
$ 74 a^{2} + 33 a + 24 + \left(94 a^{2} + 4 a + 51\right)\cdot 97 + \left(27 a^{2} + 67 a + 38\right)\cdot 97^{2} + \left(53 a^{2} + 89 a + 93\right)\cdot 97^{3} + \left(18 a^{2} + 37 a + 78\right)\cdot 97^{4} + \left(23 a^{2} + 34 a + 9\right)\cdot 97^{5} + \left(87 a^{2} + 6 a + 6\right)\cdot 97^{6} + \left(12 a^{2} + 31 a + 45\right)\cdot 97^{7} + \left(6 a^{2} + 84 a + 4\right)\cdot 97^{8} + \left(15 a^{2} + 63 a + 58\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 60 a^{2} + 74 a + 37 + \left(73 a^{2} + 4 a + 21\right)\cdot 97 + \left(13 a^{2} + 40 a + 50\right)\cdot 97^{2} + \left(92 a^{2} + 94 a + 35\right)\cdot 97^{3} + \left(85 a^{2} + 91 a + 95\right)\cdot 97^{4} + \left(48 a^{2} + 19 a + 66\right)\cdot 97^{5} + \left(12 a^{2} + 18 a + 42\right)\cdot 97^{6} + \left(89 a^{2} + 32 a + 17\right)\cdot 97^{7} + \left(80 a^{2} + 34 a + 65\right)\cdot 97^{8} + \left(31 a^{2} + 76 a + 61\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 88 a^{2} + 33 a + 11 + \left(83 a^{2} + 29 a + 83\right)\cdot 97 + \left(48 a^{2} + 19 a + 66\right)\cdot 97^{2} + \left(54 a^{2} + 13 a + 3\right)\cdot 97^{3} + \left(24 a^{2} + 30 a + 18\right)\cdot 97^{4} + \left(27 a^{2} + 79 a + 34\right)\cdot 97^{5} + \left(88 a^{2} + 19 a + 12\right)\cdot 97^{6} + \left(42 a^{2} + 73 a + 31\right)\cdot 97^{7} + \left(20 a^{2} + 2 a + 90\right)\cdot 97^{8} + \left(48 a^{2} + 43 a + 62\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 a^{2} + 26 a + 32 + \left(54 a^{2} + 64 a + 3\right)\cdot 97 + \left(52 a^{2} + 16 a + 89\right)\cdot 97^{2} + \left(76 a^{2} + 57 a + 38\right)\cdot 97^{3} + \left(80 a^{2} + 95 a + 64\right)\cdot 97^{4} + \left(18 a^{2} + 25 a + 80\right)\cdot 97^{5} + \left(61 a^{2} + 76 a + 43\right)\cdot 97^{6} + \left(61 a^{2} + 15 a + 46\right)\cdot 97^{7} + \left(75 a^{2} + 47 a + 33\right)\cdot 97^{8} + \left(74 a^{2} + 23 a + 28\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 88\cdot 97 + 31\cdot 97^{2} + 91\cdot 97^{3} + 51\cdot 97^{4} + 29\cdot 97^{5} + 45\cdot 97^{6} + 65\cdot 97^{7} + 75\cdot 97^{8} + 60\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 a^{2} + 87 a + 50 + \left(36 a^{2} + 62 a + 89\right)\cdot 97 + \left(34 a^{2} + 37 a + 76\right)\cdot 97^{2} + \left(47 a^{2} + 86 a + 57\right)\cdot 97^{3} + \left(83 a^{2} + 71 a + 80\right)\cdot 97^{4} + \left(20 a^{2} + 94 a + 92\right)\cdot 97^{5} + \left(93 a^{2} + 58 a + 41\right)\cdot 97^{6} + \left(61 a^{2} + 88 a + 48\right)\cdot 97^{7} + \left(92 a^{2} + 59 a + 38\right)\cdot 97^{8} + \left(16 a^{2} + 74 a + 69\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 + 39\cdot 97 + 2\cdot 97^{2} + 20\cdot 97^{3} + 96\cdot 97^{4} + 4\cdot 97^{5} + 21\cdot 97^{6} + 45\cdot 97^{7} + 12\cdot 97^{8} + 52\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 a^{2} + 38 a + 42 + \left(44 a^{2} + 28 a + 42\right)\cdot 97 + \left(16 a^{2} + 13 a + 66\right)\cdot 97^{2} + \left(64 a^{2} + 47 a + 61\right)\cdot 97^{3} + \left(94 a^{2} + 60 a + 50\right)\cdot 97^{4} + \left(54 a^{2} + 36 a + 6\right)\cdot 97^{5} + \left(45 a^{2} + 14 a + 47\right)\cdot 97^{6} + \left(22 a^{2} + 50 a + 5\right)\cdot 97^{7} + \left(15 a^{2} + 62 a + 59\right)\cdot 97^{8} + \left(7 a^{2} + 9 a + 10\right)\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 39 + 66\cdot 97 + 62\cdot 97^{2} + 82\cdot 97^{3} + 45\cdot 97^{4} + 62\cdot 97^{5} + 30\cdot 97^{6} + 83\cdot 97^{7} + 8\cdot 97^{8} + 81\cdot 97^{9} +O\left(97^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,3,6)(5,7,9)$ |
| $(1,8,4)(2,3,6)(5,9,7)$ |
| $(2,9)(3,5)(4,8)(6,7)$ |
| $(1,5,2,8,9,3,4,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,2)(3,4)(5,9)(6,8)$ |
$0$ |
| $2$ |
$3$ |
$(1,8,4)(2,3,6)(5,9,7)$ |
$-3$ |
| $3$ |
$3$ |
$(1,8,4)(2,6,3)$ |
$0$ |
| $3$ |
$3$ |
$(1,4,8)(2,3,6)$ |
$0$ |
| $9$ |
$6$ |
$(1,3,8,2,4,6)(5,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,6,4,2,8,3)(5,9)$ |
$0$ |
| $6$ |
$9$ |
$(1,5,2,8,9,3,4,7,6)$ |
$0$ |
| $6$ |
$9$ |
$(1,2,9,4,6,5,8,3,7)$ |
$0$ |
| $6$ |
$9$ |
$(1,7,2,8,5,3,4,9,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
