Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 3\cdot 29 + 28\cdot 29^{2} + 21\cdot 29^{3} + 7\cdot 29^{4} + 2\cdot 29^{5} + 8\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 16\cdot 29 + 8\cdot 29^{2} + 20\cdot 29^{3} + 8\cdot 29^{4} + 5\cdot 29^{5} + 6\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 24 + \left(5 a + 18\right)\cdot 29 + \left(27 a + 24\right)\cdot 29^{2} + \left(7 a + 25\right)\cdot 29^{3} + \left(9 a + 27\right)\cdot 29^{4} + \left(23 a + 3\right)\cdot 29^{5} + \left(18 a + 18\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a + 13 + \left(5 a + 5\right)\cdot 29 + \left(27 a + 15\right)\cdot 29^{2} + \left(7 a + 27\right)\cdot 29^{3} + \left(9 a + 26\right)\cdot 29^{4} + 23 a\cdot 29^{5} + \left(18 a + 20\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 3 + \left(23 a + 28\right)\cdot 29 + \left(a + 9\right)\cdot 29^{2} + \left(21 a + 9\right)\cdot 29^{3} + \left(19 a + 8\right)\cdot 29^{4} + \left(5 a + 24\right)\cdot 29^{5} + \left(10 a + 1\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 21 + \left(23 a + 14\right)\cdot 29 + a\cdot 29^{2} + \left(21 a + 11\right)\cdot 29^{3} + \left(19 a + 7\right)\cdot 29^{4} + \left(5 a + 21\right)\cdot 29^{5} + \left(10 a + 3\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)(2,4)(5,6)$ |
| $(3,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,4)(2,3)$ |
$0$ |
| $2$ |
$3$ |
$(1,4,6)(2,3,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,5,4,2,6,3)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
