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