Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 5\cdot 29 + 25\cdot 29^{2} + 21\cdot 29^{3} + 13\cdot 29^{4} + 24\cdot 29^{5} + 19\cdot 29^{6} + 4\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 19\cdot 29 + 8\cdot 29^{2} + 22\cdot 29^{3} + 8\cdot 29^{4} + 24\cdot 29^{5} + 24\cdot 29^{6} + 14\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a^{2} + 13 a + 2 + \left(28 a + 19\right)\cdot 29 + \left(4 a^{2} + 3 a + 10\right)\cdot 29^{2} + \left(19 a^{2} + 13 a + 25\right)\cdot 29^{3} + \left(20 a^{2} + 11 a + 8\right)\cdot 29^{4} + \left(11 a^{2} + 3 a + 23\right)\cdot 29^{5} + \left(16 a^{2} + 18 a + 28\right)\cdot 29^{6} + \left(6 a^{2} + 9 a + 12\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a^{2} + 24 a + 12 + \left(18 a^{2} + 28 a + 21\right)\cdot 29 + \left(14 a^{2} + 19 a + 9\right)\cdot 29^{2} + \left(18 a^{2} + 27 a\right)\cdot 29^{3} + \left(5 a^{2} + 21 a + 16\right)\cdot 29^{4} + \left(6 a^{2} + 24 a + 7\right)\cdot 29^{5} + \left(17 a^{2} + 3 a + 8\right)\cdot 29^{6} + \left(15 a^{2} + 6 a + 22\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 22 + \left(9 a^{2} + 6 a + 20\right)\cdot 29 + \left(7 a^{2} + 17 a + 24\right)\cdot 29^{2} + \left(17 a^{2} + 23 a + 22\right)\cdot 29^{3} + \left(7 a^{2} + a + 10\right)\cdot 29^{4} + \left(8 a^{2} + 3 a + 28\right)\cdot 29^{5} + \left(24 a^{2} + 13 a + 19\right)\cdot 29^{6} + \left(17 a^{2} + 23 a + 8\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 + 18\cdot 29 + 8\cdot 29^{2} + 8\cdot 29^{4} + 17\cdot 29^{5} + 7\cdot 29^{6} + 21\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 21 a^{2} + 25 a + 15 + \left(10 a + 16\right)\cdot 29 + \left(27 a^{2} + 8 a + 16\right)\cdot 29^{2} + \left(13 a + 15\right)\cdot 29^{3} + \left(28 a^{2} + 28 a + 26\right)\cdot 29^{4} + \left(3 a^{2} + 21 a + 23\right)\cdot 29^{5} + \left(28 a^{2} + 5 a + 22\right)\cdot 29^{6} + \left(11 a^{2} + 7\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 11 a^{2} + 9 a + 21 + \left(9 a^{2} + 18 a + 8\right)\cdot 29 + \left(16 a^{2} + 2\right)\cdot 29^{2} + \left(9 a^{2} + 17 a + 27\right)\cdot 29^{3} + \left(24 a^{2} + 7 a + 11\right)\cdot 29^{4} + \left(18 a^{2} + 11 a + 24\right)\cdot 29^{5} + \left(12 a^{2} + 19 a + 11\right)\cdot 29^{6} + \left(a^{2} + 22 a + 3\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 15 a^{2} + 13 + \left(19 a^{2} + 23 a + 15\right)\cdot 29 + \left(17 a^{2} + 7 a + 9\right)\cdot 29^{2} + \left(21 a^{2} + 21 a + 9\right)\cdot 29^{3} + \left(15 a + 11\right)\cdot 29^{4} + \left(9 a^{2} + 22 a\right)\cdot 29^{5} + \left(17 a^{2} + 26 a + 1\right)\cdot 29^{6} + \left(4 a^{2} + 24 a + 20\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,6,2)(4,8,7)$ |
| $(1,3,4,6,5,7,2,9,8)$ |
| $(3,5,9)(4,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $1$ |
$3$ |
$(1,6,2)(3,5,9)(4,7,8)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $1$ |
$3$ |
$(1,2,6)(3,9,5)(4,8,7)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $3$ |
$3$ |
$(1,6,2)(4,8,7)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,2,6)(4,7,8)$ |
$0$ |
$0$ |
| $3$ |
$9$ |
$(1,3,4,6,5,7,2,9,8)$ |
$0$ |
$0$ |
| $3$ |
$9$ |
$(1,4,5,2,8,3,6,7,9)$ |
$0$ |
$0$ |
| $3$ |
$9$ |
$(1,5,4,6,9,7,2,3,8)$ |
$0$ |
$0$ |
| $3$ |
$9$ |
$(1,4,9,2,8,5,6,7,3)$ |
$0$ |
$0$ |
| $3$ |
$9$ |
$(1,9,4,6,3,7,2,5,8)$ |
$0$ |
$0$ |
| $3$ |
$9$ |
$(1,4,3,2,8,9,6,7,5)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
