Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 11 + 24\cdot 29 + \left(24 a + 12\right)\cdot 29^{2} + \left(18 a + 4\right)\cdot 29^{3} + \left(28 a + 3\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 1 + \left(13 a + 6\right)\cdot 29 + \left(7 a + 4\right)\cdot 29^{2} + \left(23 a + 5\right)\cdot 29^{3} + \left(16 a + 15\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 13 + \left(28 a + 14\right)\cdot 29 + \left(4 a + 16\right)\cdot 29^{2} + \left(10 a + 16\right)\cdot 29^{3} + 11\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 17 + \left(15 a + 5\right)\cdot 29 + \left(21 a + 28\right)\cdot 29^{2} + \left(5 a + 26\right)\cdot 29^{3} + \left(12 a + 17\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 a + 9 + \left(27 a + 18\right)\cdot 29 + \left(11 a + 25\right)\cdot 29^{2} + \left(5 a + 23\right)\cdot 29^{3} + \left(28 a + 9\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 8 + \left(a + 18\right)\cdot 29 + \left(17 a + 28\right)\cdot 29^{2} + \left(23 a + 9\right)\cdot 29^{3} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2,5)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,2)(3,6,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,4,5,3,2,6)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,6,2,3,5,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
