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