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:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 14 + \left(21 a + 4\right)\cdot 29 + \left(9 a + 15\right)\cdot 29^{2} + \left(28 a + 20\right)\cdot 29^{3} + \left(28 a + 10\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 28 + 21 a\cdot 29 + 9 a\cdot 29^{2} + \left(28 a + 22\right)\cdot 29^{3} + \left(28 a + 15\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 8 + \left(7 a + 27\right)\cdot 29 + \left(19 a + 26\right)\cdot 29^{2} + 8\cdot 29^{3} + 16\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 3 + \left(21 a + 13\right)\cdot 29 + \left(9 a + 2\right)\cdot 29^{2} + \left(28 a + 6\right)\cdot 29^{3} + \left(28 a + 16\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 23 + \left(7 a + 1\right)\cdot 29 + \left(19 a + 13\right)\cdot 29^{2} + 7\cdot 29^{3} + 11\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 12 + \left(7 a + 10\right)\cdot 29 + 19 a\cdot 29^{2} + 22\cdot 29^{3} + 16\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,2)(3,5,6)$ |
| $(1,5)(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,4,2)(3,5,6)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$3$ |
$(1,2,4)(3,6,5)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$6$ |
$(1,6,2,5,4,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,4,5,2,6)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
