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 + 5\cdot 29 + 29^{2} + 13\cdot 29^{3} + 4\cdot 29^{4} + 2\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 11 + \left(11 a + 17\right)\cdot 29 + \left(19 a + 16\right)\cdot 29^{2} + \left(17 a + 12\right)\cdot 29^{3} + \left(5 a + 16\right)\cdot 29^{4} + \left(28 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 10\cdot 29 + 21\cdot 29^{2} + 22\cdot 29^{3} + 28\cdot 29^{4} + 4\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 16 + \left(17 a + 3\right)\cdot 29 + \left(4 a + 28\right)\cdot 29^{2} + \left(10 a + 27\right)\cdot 29^{3} + \left(17 a + 24\right)\cdot 29^{4} + \left(7 a + 3\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 17 + \left(17 a + 8\right)\cdot 29 + \left(9 a + 15\right)\cdot 29^{2} + \left(11 a + 23\right)\cdot 29^{3} + \left(23 a + 26\right)\cdot 29^{4} + 20\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 10 + \left(11 a + 12\right)\cdot 29 + \left(24 a + 4\right)\cdot 29^{2} + \left(18 a + 16\right)\cdot 29^{3} + \left(11 a + 14\right)\cdot 29^{4} + \left(21 a + 24\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,5)$ |
| $(1,6,4)$ |
| $(1,3,6,5,4,2)$ |
| $(3,5)(4,6)$ |
| $(1,6)(2,3)$ |
| $(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $3$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$0$ |
| $3$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$0$ |
| $9$ |
$2$ |
$(1,6)(2,3)$ |
$0$ |
| $2$ |
$3$ |
$(1,6,4)(2,3,5)$ |
$-2$ |
| $2$ |
$3$ |
$(1,4,6)(2,3,5)$ |
$-2$ |
| $4$ |
$3$ |
$(1,6,4)$ |
$1$ |
| $6$ |
$6$ |
$(1,3,6,5,4,2)$ |
$0$ |
| $6$ |
$6$ |
$(1,3,4,5,6,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
