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:
| $r_{ 1 }$ |
$=$ |
$ 27 a^{2} + 36 a + 47 + \left(65 a^{2} + 95 a + 71\right)\cdot 97 + \left(61 a^{2} + 3 a + 32\right)\cdot 97^{2} + \left(29 a^{2} + 14 a + 27\right)\cdot 97^{3} + \left(37 a^{2} + 2 a + 40\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a^{2} + 5 a + 73 + \left(95 a^{2} + 77 a + 87\right)\cdot 97 + \left(79 a^{2} + 15 a\right)\cdot 97^{2} + \left(64 a^{2} + 43 a + 59\right)\cdot 97^{3} + \left(20 a^{2} + 75 a\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 94 a^{2} + 65 a + 19 + \left(96 a^{2} + 9 a + 2\right)\cdot 97 + \left(37 a^{2} + 77 a + 40\right)\cdot 97^{2} + \left(34 a^{2} + a + 70\right)\cdot 97^{3} + \left(89 a^{2} + 2 a + 24\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 65 a^{2} + 67 a + 81 + \left(55 a^{2} + 26 a + 13\right)\cdot 97 + \left(a^{2} + 77 a + 60\right)\cdot 97^{2} + \left(5 a^{2} + 2 a + 73\right)\cdot 97^{3} + \left(26 a^{2} + 23 a + 69\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 78 a^{2} + 45 a + 30 + \left(68 a^{2} + 7 a + 19\right)\cdot 97 + \left(38 a^{2} + 70 a + 46\right)\cdot 97^{2} + \left(43 a^{2} + 65 a + 29\right)\cdot 97^{3} + \left(9 a^{2} + 56 a + 40\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a^{2} + 37 a + 15 + \left(53 a^{2} + 23 a + 23\right)\cdot 97 + \left(68 a^{2} + 18 a + 31\right)\cdot 97^{2} + \left(a^{2} + 57 a + 70\right)\cdot 97^{3} + \left(42 a^{2} + 33 a + 41\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 89 a^{2} + 15 a + 96 + \left(71 a^{2} + 66 a + 37\right)\cdot 97 + \left(86 a^{2} + 8 a + 43\right)\cdot 97^{2} + \left(51 a^{2} + 71 a + 80\right)\cdot 97^{3} + \left(45 a^{2} + 6 a + 62\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 94 a^{2} + 27 a + 19 + \left(a^{2} + 10 a + 14\right)\cdot 97 + \left(76 a^{2} + 4 a + 74\right)\cdot 97^{2} + \left(94 a^{2} + 52 a + 44\right)\cdot 97^{3} + \left(83 a^{2} + 19 a + 89\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 5 a^{2} + 91 a + 12 + \left(73 a^{2} + 71 a + 21\right)\cdot 97 + \left(33 a^{2} + 15 a + 59\right)\cdot 97^{2} + \left(62 a^{2} + 80 a + 29\right)\cdot 97^{3} + \left(33 a^{2} + 71 a + 18\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,6,2,9,7,3,4,5,8)$ |
| $(1,9,4)(2,3,8)(5,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $3$ | $(1,9,4)(2,3,8)(5,6,7)$ | $-1$ |
| $2$ | $9$ | $(1,6,2,9,7,3,4,5,8)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,2,7,4,8,6,9,3,5)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,7,8,9,5,2,4,6,3)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
