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:
| $r_{ 1 }$ |
$=$ |
$ 43 a^{2} + 49 a + 3 + \left(25 a^{2} + 74 a + 51\right)\cdot 83 + \left(29 a^{2} + 3 a + 58\right)\cdot 83^{2} + \left(8 a^{2} + 22 a + 16\right)\cdot 83^{3} + 37 a\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 a^{2} + 80 a + 29 + \left(55 a^{2} + 28 a + 28\right)\cdot 83 + \left(62 a^{2} + 81 a + 42\right)\cdot 83^{2} + \left(71 a^{2} + 30 a + 60\right)\cdot 83^{3} + \left(3 a^{2} + 64 a + 7\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 71 a^{2} + 52 a + 59 + \left(36 a^{2} + 18 a + 73\right)\cdot 83 + \left(10 a^{2} + 77 a + 20\right)\cdot 83^{2} + \left(56 a^{2} + 81 a + 29\right)\cdot 83^{3} + \left(49 a^{2} + 52 a + 16\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a^{2} + 37 a + 20 + \left(23 a^{2} + 24 a + 46\right)\cdot 83 + \left(4 a^{2} + 22 a + 8\right)\cdot 83^{2} + \left(80 a^{2} + 50 a + 77\right)\cdot 83^{3} + \left(8 a^{2} + 34 a + 17\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a^{2} + 49 a + 34 + \left(4 a^{2} + 29 a + 8\right)\cdot 83 + \left(16 a^{2} + 62 a + 32\right)\cdot 83^{2} + \left(14 a^{2} + a + 28\right)\cdot 83^{3} + \left(70 a^{2} + 67 a + 57\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 a^{2} + 63 a + 23 + \left(20 a^{2} + a + 41\right)\cdot 83 + \left(29 a^{2} + 57 a + 58\right)\cdot 83^{2} + 70 a\cdot 83^{3} + \left(3 a^{2} + 25 a + 6\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 52 a^{2} + 65 a + 21 + \left(20 a^{2} + 72 a + 41\right)\cdot 83 + \left(43 a^{2} + a + 3\right)\cdot 83^{2} + \left(18 a^{2} + 62 a + 37\right)\cdot 83^{3} + \left(33 a^{2} + 75 a + 66\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 13 a^{2} + 20 a + 26 + \left(81 a^{2} + 42 a + 79\right)\cdot 83 + \left(31 a^{2} + 77 a + 63\right)\cdot 83^{2} + \left(66 a^{2} + 58 a + 49\right)\cdot 83^{3} + \left(76 a^{2} + 72 a + 70\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 17 a^{2} + 34 + \left(64 a^{2} + 39 a + 45\right)\cdot 83 + \left(21 a^{2} + 31 a + 43\right)\cdot 83^{2} + \left(16 a^{2} + 36 a + 32\right)\cdot 83^{3} + \left(3 a^{2} + 67 a + 6\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,7)(2,4,5)(6,9,8)$ |
| $(1,2,8,7,5,9,3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
$c5$ |
$c6$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$3$ |
$(1,7,3)(2,5,4)(6,8,9)$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
| $1$ |
$3$ |
$(1,3,7)(2,4,5)(6,9,8)$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
| $1$ |
$9$ |
$(1,2,8,7,5,9,3,4,6)$ |
$\zeta_{9}$ |
$\zeta_{9}^{2}$ |
$\zeta_{9}^{4}$ |
$\zeta_{9}^{5}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
| $1$ |
$9$ |
$(1,8,5,3,6,2,7,9,4)$ |
$\zeta_{9}^{2}$ |
$\zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$\zeta_{9}$ |
$\zeta_{9}^{5}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
| $1$ |
$9$ |
$(1,5,6,7,4,8,3,2,9)$ |
$\zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$\zeta_{9}^{2}$ |
$\zeta_{9}$ |
$\zeta_{9}^{5}$ |
| $1$ |
$9$ |
$(1,9,2,3,8,4,7,6,5)$ |
$\zeta_{9}^{5}$ |
$\zeta_{9}$ |
$\zeta_{9}^{2}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$\zeta_{9}^{4}$ |
| $1$ |
$9$ |
$(1,4,9,7,2,6,3,5,8)$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$\zeta_{9}^{5}$ |
$\zeta_{9}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$\zeta_{9}^{4}$ |
$\zeta_{9}^{2}$ |
| $1$ |
$9$ |
$(1,6,4,3,9,5,7,8,2)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$\zeta_{9}^{5}$ |
$\zeta_{9}^{4}$ |
$\zeta_{9}^{2}$ |
$\zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
