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:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 6 + \left(26 a + 16\right)\cdot 29 + \left(15 a + 22\right)\cdot 29^{2} + \left(13 a + 13\right)\cdot 29^{3} + 20\cdot 29^{4} + \left(12 a + 3\right)\cdot 29^{5} + \left(9 a + 24\right)\cdot 29^{6} + \left(21 a + 12\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 10 + \left(14 a + 13\right)\cdot 29 + \left(16 a + 28\right)\cdot 29^{2} + \left(26 a + 6\right)\cdot 29^{3} + \left(15 a + 4\right)\cdot 29^{4} + \left(6 a + 17\right)\cdot 29^{5} + \left(17 a + 28\right)\cdot 29^{6} + \left(15 a + 19\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 27 + \left(14 a + 12\right)\cdot 29 + \left(12 a + 9\right)\cdot 29^{2} + \left(2 a + 7\right)\cdot 29^{3} + \left(13 a + 28\right)\cdot 29^{4} + \left(22 a + 4\right)\cdot 29^{5} + \left(11 a + 21\right)\cdot 29^{6} + \left(13 a + 22\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 11 a\cdot 29 + \left(a + 21\right)\cdot 29^{2} + \left(28 a + 28\right)\cdot 29^{3} + \left(26 a + 16\right)\cdot 29^{4} + \left(2 a + 19\right)\cdot 29^{5} + \left(a + 4\right)\cdot 29^{6} + \left(5 a + 21\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 21 + \left(17 a + 17\right)\cdot 29 + \left(27 a + 16\right)\cdot 29^{2} + 22\cdot 29^{3} + \left(2 a + 7\right)\cdot 29^{4} + \left(26 a + 7\right)\cdot 29^{5} + \left(27 a + 7\right)\cdot 29^{6} + \left(23 a + 16\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 26 + \left(2 a + 26\right)\cdot 29 + \left(13 a + 17\right)\cdot 29^{2} + \left(15 a + 7\right)\cdot 29^{3} + \left(28 a + 9\right)\cdot 29^{4} + \left(16 a + 5\right)\cdot 29^{5} + \left(19 a + 1\right)\cdot 29^{6} + \left(7 a + 23\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,5)(3,4,6)$ |
| $(1,2,5)$ |
| $(1,3,5,6,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,2,5)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,2)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,2)(3,4,6)$ | $-1$ |
| $3$ | $6$ | $(1,3,5,6,2,4)$ | $0$ |
| $3$ | $6$ | $(1,4,2,6,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
