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