Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 6 + \left(21 a + 1\right)\cdot 29 + \left(20 a + 6\right)\cdot 29^{2} + \left(24 a + 10\right)\cdot 29^{3} + \left(28 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 23 + \left(10 a + 9\right)\cdot 29 + \left(23 a + 6\right)\cdot 29^{2} + \left(15 a + 13\right)\cdot 29^{3} + \left(26 a + 1\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 + 9\cdot 29 + 26\cdot 29^{2} + 4\cdot 29^{3} + 25\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 3 + \left(18 a + 10\right)\cdot 29 + \left(5 a + 25\right)\cdot 29^{2} + \left(13 a + 10\right)\cdot 29^{3} + \left(2 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 14 + \left(7 a + 3\right)\cdot 29 + \left(8 a + 1\right)\cdot 29^{2} + \left(4 a + 26\right)\cdot 29^{3} + 5\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 + 24\cdot 29 + 21\cdot 29^{2} + 21\cdot 29^{3} + 20\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,5)$ |
| $(1,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $-2$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,5,6)$ | $1$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
| $12$ | $6$ | $(1,5,6)(3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
