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