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