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 + 26 + \left(6 a + 5\right)\cdot 29 + \left(25 a + 23\right)\cdot 29^{2} + \left(17 a + 23\right)\cdot 29^{3} + \left(26 a + 14\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 23\cdot 29 + 13\cdot 29^{2} + 11\cdot 29^{3} + 3\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 23 + \left(26 a + 21\right)\cdot 29 + \left(25 a + 1\right)\cdot 29^{2} + \left(21 a + 10\right)\cdot 29^{3} + \left(26 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 4 + \left(2 a + 5\right)\cdot 29 + \left(3 a + 18\right)\cdot 29^{2} + \left(7 a + 6\right)\cdot 29^{3} + \left(2 a + 18\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 5 + \left(22 a + 20\right)\cdot 29 + \left(3 a + 26\right)\cdot 29^{2} + 11 a\cdot 29^{3} + \left(2 a + 14\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 10\cdot 29 + 3\cdot 29^{2} + 5\cdot 29^{3} + 14\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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(2,3)$ | $-2$ |
| $9$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $-2$ |
| $4$ | $3$ | $(2,3,4)$ | $1$ |
| $18$ | $4$ | $(1,2,5,3)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,2,5,3,6,4)$ | $0$ |
| $12$ | $6$ | $(1,5,6)(2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
