Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{3} + 3 x + 81 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 56 a^{2} + 14 a + 81 + \left(60 a^{2} + 24 a + 79\right)\cdot 83 + \left(67 a^{2} + 72 a + 44\right)\cdot 83^{2} + \left(80 a^{2} + 67 a + 58\right)\cdot 83^{3} + \left(28 a^{2} + 34 a + 20\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a^{2} + 7 a + 29 + \left(76 a^{2} + 33 a + 28\right)\cdot 83 + \left(32 a^{2} + 58\right)\cdot 83^{2} + \left(29 a^{2} + 25 a + 38\right)\cdot 83^{3} + \left(60 a^{2} + a\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a^{2} + 60 a + 33 + \left(15 a^{2} + 79 a + 62\right)\cdot 83 + \left(28 a^{2} + 60 a + 32\right)\cdot 83^{2} + \left(70 a^{2} + 72 a + 31\right)\cdot 83^{3} + \left(42 a^{2} + 65 a + 20\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a^{2} + 34 a + 66 + \left(34 a^{2} + 47 a + 78\right)\cdot 83 + \left(48 a^{2} + 15 a + 44\right)\cdot 83^{2} + \left(6 a^{2} + 68 a + 59\right)\cdot 83^{3} + \left(20 a^{2} + 40 a + 59\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a^{2} + 52 a + 29 + \left(23 a^{2} + 19 a + 78\right)\cdot 83 + \left(10 a^{2} + 62 a + 79\right)\cdot 83^{2} + \left(72 a^{2} + 68 a + 34\right)\cdot 83^{3} + \left(78 a^{2} + 40 a + 9\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a^{2} + 81 a + 37 + \left(51 a^{2} + 45 a + 28\right)\cdot 83 + \left(75 a^{2} + 81 a + 16\right)\cdot 83^{2} + \left(73 a^{2} + a + 28\right)\cdot 83^{3} + \left(45 a^{2} + 33 a + 28\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 80 a^{2} + 62 a + 46 + \left(28 a^{2} + 25 a + 16\right)\cdot 83 + \left(65 a^{2} + 10 a + 40\right)\cdot 83^{2} + \left(55 a^{2} + 73 a + 8\right)\cdot 83^{3} + \left(76 a^{2} + 46 a + 33\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 25 a^{2} + 51 a + 2 + \left(80 a^{2} + 72 a + 4\right)\cdot 83 + \left(41 a^{2} + 68 a + 32\right)\cdot 83^{2} + \left(2 a^{2} + 12 a + 51\right)\cdot 83^{3} + \left(17 a^{2} + 9 a + 53\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 49 a^{2} + 54 a + 12 + \left(44 a^{2} + 66 a + 38\right)\cdot 83 + \left(44 a^{2} + 42 a + 65\right)\cdot 83^{2} + \left(23 a^{2} + 24 a + 20\right)\cdot 83^{3} + \left(44 a^{2} + 59 a + 23\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,4,7,9,6,2,5,8)$ |
| $(1,7,2)(3,9,5)(4,6,8)$ |
| $(1,5)(2,3)(4,6)(7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,5)(2,3)(4,6)(7,9)$ | $0$ |
| $2$ | $3$ | $(1,7,2)(3,9,5)(4,6,8)$ | $-1$ |
| $2$ | $9$ | $(1,3,4,7,9,6,2,5,8)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,4,9,2,8,3,7,6,5)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,9,8,7,5,4,2,3,6)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
