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:
| $r_{ 1 }$ |
$=$ |
$ 23 a + 9 + \left(24 a + 24\right)\cdot 29 + \left(5 a + 8\right)\cdot 29^{2} + \left(2 a + 7\right)\cdot 29^{3} + \left(5 a + 12\right)\cdot 29^{4} + \left(12 a + 17\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 4\cdot 29 + 18\cdot 29^{2} + 23\cdot 29^{3} + 9\cdot 29^{4} + 7\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 19\cdot 29 + 23\cdot 29^{2} + 7\cdot 29^{3} + 4\cdot 29^{4} + 10\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 19 + 4 a\cdot 29 + \left(18 a + 12\right)\cdot 29^{2} + \left(13 a + 7\right)\cdot 29^{3} + \left(27 a + 23\right)\cdot 29^{4} + \left(5 a + 2\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 28 + \left(24 a + 28\right)\cdot 29 + \left(10 a + 10\right)\cdot 29^{2} + \left(15 a + 28\right)\cdot 29^{3} + \left(a + 1\right)\cdot 29^{4} + \left(23 a + 5\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 8 + \left(4 a + 9\right)\cdot 29 + \left(23 a + 13\right)\cdot 29^{2} + \left(26 a + 12\right)\cdot 29^{3} + \left(23 a + 6\right)\cdot 29^{4} + \left(16 a + 15\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,2)(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,3,6)(2,5,4)$ | $-1$ |
| $2$ | $6$ | $(1,5,3,4,6,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
