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