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