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