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^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 13\cdot 29 + 26\cdot 29^{2} + 17\cdot 29^{3} + 7\cdot 29^{4} + 5\cdot 29^{5} + 3\cdot 29^{6} + 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 15\cdot 29 + 2\cdot 29^{2} + 11\cdot 29^{3} + 21\cdot 29^{4} + 23\cdot 29^{5} + 25\cdot 29^{6} + 27\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 1 + \left(3 a + 27\right)\cdot 29 + \left(3 a + 20\right)\cdot 29^{2} + \left(6 a + 13\right)\cdot 29^{3} + \left(5 a + 27\right)\cdot 29^{4} + \left(24 a + 11\right)\cdot 29^{5} + \left(17 a + 5\right)\cdot 29^{6} + \left(2 a + 24\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 9 + \left(25 a + 26\right)\cdot 29 + \left(25 a + 3\right)\cdot 29^{2} + \left(22 a + 12\right)\cdot 29^{3} + \left(23 a + 18\right)\cdot 29^{4} + \left(4 a + 11\right)\cdot 29^{5} + \left(11 a + 12\right)\cdot 29^{6} + \left(26 a + 19\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + \left(25 a + 2\right)\cdot 29 + \left(25 a + 8\right)\cdot 29^{2} + \left(22 a + 15\right)\cdot 29^{3} + \left(23 a + 1\right)\cdot 29^{4} + \left(4 a + 17\right)\cdot 29^{5} + \left(11 a + 23\right)\cdot 29^{6} + \left(26 a + 4\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 a + 21 + \left(3 a + 2\right)\cdot 29 + \left(3 a + 25\right)\cdot 29^{2} + \left(6 a + 16\right)\cdot 29^{3} + \left(5 a + 10\right)\cdot 29^{4} + \left(24 a + 17\right)\cdot 29^{5} + \left(17 a + 16\right)\cdot 29^{6} + \left(2 a + 9\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,3,4)(2,5,6)$ |
| $(1,3)(2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $-1$ |
| $6$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,3,4)(2,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,2,3)$ | $-1$ |
| $6$ | $4$ | $(1,2)(3,6,5,4)$ | $1$ |
| $8$ | $6$ | $(1,5,6,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
