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