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:
| $r_{ 1 }$ |
$=$ |
$ 16 a + 22 + 21 a\cdot 29 + \left(18 a + 7\right)\cdot 29^{2} + \left(12 a + 6\right)\cdot 29^{3} + \left(23 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a + 11 + \left(21 a + 9\right)\cdot 29 + \left(18 a + 23\right)\cdot 29^{2} + \left(12 a + 20\right)\cdot 29^{3} + \left(23 a + 7\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 4 + \left(7 a + 14\right)\cdot 29 + \left(10 a + 8\right)\cdot 29^{2} + \left(16 a + 7\right)\cdot 29^{3} + \left(5 a + 25\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + \left(7 a + 2\right)\cdot 29 + \left(10 a + 6\right)\cdot 29^{2} + \left(16 a + 23\right)\cdot 29^{3} + \left(5 a + 24\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 7 + \left(21 a + 26\right)\cdot 29 + \left(18 a + 20\right)\cdot 29^{2} + \left(12 a + 7\right)\cdot 29^{3} + \left(23 a + 7\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 15 + \left(7 a + 5\right)\cdot 29 + \left(10 a + 21\right)\cdot 29^{2} + \left(16 a + 21\right)\cdot 29^{3} + \left(5 a + 19\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,3)(4,5)$ |
| $(1,2,5)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,2)(3,6,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,5,6,2,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,2,6,5,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
