Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{3} + 9 x + 92 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 a^{2} + 92 a + 2 + \left(9 a^{2} + 75 a + 47\right)\cdot 97 + \left(75 a^{2} + 73 a + 93\right)\cdot 97^{2} + \left(31 a^{2} + 59 a + 17\right)\cdot 97^{3} + \left(45 a^{2} + 59 a + 54\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 93 a^{2} + 64 a + 50 + \left(11 a^{2} + 63 a + 27\right)\cdot 97 + \left(13 a^{2} + 16 a + 6\right)\cdot 97^{2} + \left(63 a^{2} + 48 a + 76\right)\cdot 97^{3} + \left(76 a^{2} + 40 a + 70\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 a^{2} + 39 a + 80 + \left(41 a^{2} + 77 a + 45\right)\cdot 97 + \left(59 a^{2} + 30 a + 96\right)\cdot 97^{2} + \left(2 a^{2} + 21 a + 36\right)\cdot 97^{3} + \left(86 a^{2} + 80 a + 7\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a^{2} + 33 a + 32 + \left(91 a^{2} + 4 a + 17\right)\cdot 97 + \left(46 a^{2} + 94 a + 32\right)\cdot 97^{2} + \left(11 a^{2} + 72 a + 59\right)\cdot 97^{3} + \left(16 a^{2} + 84 a + 24\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 a^{2} + 85 a + 3 + \left(70 a^{2} + 89 a + 89\right)\cdot 97 + \left(13 a^{2} + 49 a + 9\right)\cdot 97^{2} + \left(88 a^{2} + 43 a + 32\right)\cdot 97^{3} + \left(96 a^{2} + 53 a + 95\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a^{2} + 72 a + 36 + \left(47 a^{2} + 32 a + 46\right)\cdot 97 + \left(69 a^{2} + 4 a + 70\right)\cdot 97^{2} + \left(52 a^{2} + 58 a + 15\right)\cdot 97^{3} + \left(14 a^{2} + 19 a + 15\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 a^{2} + 63 a + 9 + \left(46 a^{2} + 40 a + 74\right)\cdot 97 + \left(59 a^{2} + 89 a + 96\right)\cdot 97^{2} + \left(62 a^{2} + 15 a + 8\right)\cdot 97^{3} + \left(62 a^{2} + 54 a + 61\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 62 a^{2} + 89 a + 10 + \left(55 a^{2} + 59 a + 96\right)\cdot 97 + \left(77 a^{2} + 95 a + 21\right)\cdot 97^{2} + \left(32 a^{2} + 62 a + 90\right)\cdot 97^{3} + \left(66 a^{2} + 89 a + 34\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 32 a^{2} + 45 a + 72 + \left(14 a^{2} + 40 a + 41\right)\cdot 97 + \left(70 a^{2} + 30 a + 57\right)\cdot 97^{2} + \left(42 a^{2} + 5 a + 50\right)\cdot 97^{3} + \left(20 a^{2} + 3 a + 24\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,8,3,5,6,7,9,4)$ |
| $(1,3,7)(2,5,9)(4,8,6)$ |
| $(1,3)(2,8)(4,5)(6,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $9$ |
$2$ |
$(1,3)(2,8)(4,5)(6,9)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$3$ |
$(1,3,7)(2,5,9)(4,8,6)$ |
$-1$ |
$-1$ |
$-1$ |
| $2$ |
$9$ |
$(1,2,8,3,5,6,7,9,4)$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ |
$9$ |
$(1,8,5,7,4,2,3,6,9)$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ |
$9$ |
$(1,5,4,3,9,8,7,2,6)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
