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:
| $r_{ 1 }$ |
$=$ |
$ 24 + 26\cdot 29 + 7\cdot 29^{2} + 8\cdot 29^{3} + 2\cdot 29^{4} + 15\cdot 29^{5} + 15\cdot 29^{6} + 24\cdot 29^{7} + 11\cdot 29^{8} + 14\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a + 17 + 11\cdot 29 + \left(15 a + 6\right)\cdot 29^{2} + \left(a + 18\right)\cdot 29^{3} + \left(10 a + 4\right)\cdot 29^{4} + \left(20 a + 12\right)\cdot 29^{5} + \left(20 a + 16\right)\cdot 29^{6} + \left(27 a + 13\right)\cdot 29^{7} + \left(10 a + 15\right)\cdot 29^{8} + \left(6 a + 18\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 10 + \left(21 a + 1\right)\cdot 29 + \left(8 a + 18\right)\cdot 29^{2} + \left(10 a + 7\right)\cdot 29^{3} + \left(21 a + 24\right)\cdot 29^{4} + \left(25 a + 18\right)\cdot 29^{5} + \left(a + 22\right)\cdot 29^{6} + \left(23 a + 15\right)\cdot 29^{7} + \left(16 a + 27\right)\cdot 29^{8} + \left(13 a + 17\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 + 2\cdot 29 + 21\cdot 29^{2} + 20\cdot 29^{3} + 26\cdot 29^{4} + 13\cdot 29^{5} + 13\cdot 29^{6} + 4\cdot 29^{7} + 17\cdot 29^{8} + 14\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a + 12 + \left(28 a + 17\right)\cdot 29 + \left(13 a + 22\right)\cdot 29^{2} + \left(27 a + 10\right)\cdot 29^{3} + \left(18 a + 24\right)\cdot 29^{4} + \left(8 a + 16\right)\cdot 29^{5} + \left(8 a + 12\right)\cdot 29^{6} + \left(a + 15\right)\cdot 29^{7} + \left(18 a + 13\right)\cdot 29^{8} + \left(22 a + 10\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 19 + \left(7 a + 27\right)\cdot 29 + \left(20 a + 10\right)\cdot 29^{2} + \left(18 a + 21\right)\cdot 29^{3} + \left(7 a + 4\right)\cdot 29^{4} + \left(3 a + 10\right)\cdot 29^{5} + \left(27 a + 6\right)\cdot 29^{6} + \left(5 a + 13\right)\cdot 29^{7} + \left(12 a + 1\right)\cdot 29^{8} + \left(15 a + 11\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6)(2,3,4)$ |
| $(1,2)(4,5)$ |
| $(1,5)(2,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,5,6)(2,3,4)$ | $0$ |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
