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