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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 24 + 2 a\cdot 29 + \left(22 a + 3\right)\cdot 29^{2} + \left(16 a + 28\right)\cdot 29^{3} + \left(13 a + 4\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 12 + \left(26 a + 27\right)\cdot 29 + \left(6 a + 23\right)\cdot 29^{2} + \left(12 a + 2\right)\cdot 29^{3} + \left(15 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 10 + \left(2 a + 4\right)\cdot 29 + \left(22 a + 18\right)\cdot 29^{2} + \left(16 a + 26\right)\cdot 29^{3} + \left(13 a + 28\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 16 + \left(26 a + 10\right)\cdot 29 + \left(6 a + 26\right)\cdot 29^{2} + \left(12 a + 15\right)\cdot 29^{3} + \left(15 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 27 + \left(26 a + 1\right)\cdot 29 + \left(6 a + 10\right)\cdot 29^{2} + \left(12 a + 1\right)\cdot 29^{3} + \left(15 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 a + 28 + \left(2 a + 12\right)\cdot 29 + \left(22 a + 5\right)\cdot 29^{2} + \left(16 a + 12\right)\cdot 29^{3} + \left(13 a + 5\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,3,6)(2,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,6)(2,5,4)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,6,3)(2,4,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,6,2,3,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,3,2,6,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
