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:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 17 + \left(8 a + 17\right)\cdot 29 + \left(25 a + 14\right)\cdot 29^{2} + \left(24 a + 20\right)\cdot 29^{3} + \left(3 a + 3\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 28 + \left(27 a + 15\right)\cdot 29 + \left(11 a + 12\right)\cdot 29^{2} + \left(20 a + 19\right)\cdot 29^{3} + \left(21 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 6 + \left(20 a + 11\right)\cdot 29 + \left(3 a + 16\right)\cdot 29^{2} + \left(4 a + 3\right)\cdot 29^{3} + \left(25 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 1 + \left(a + 25\right)\cdot 29 + \left(17 a + 15\right)\cdot 29^{2} + \left(8 a + 22\right)\cdot 29^{3} + \left(7 a + 23\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 4 + \left(2 a + 6\right)\cdot 29 + \left(a + 12\right)\cdot 29^{2} + \left(14 a + 19\right)\cdot 29^{3} + 4 a\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 5 + \left(26 a + 11\right)\cdot 29 + \left(27 a + 15\right)\cdot 29^{2} + \left(14 a + 1\right)\cdot 29^{3} + \left(24 a + 9\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,4,3,5,2)$ |
| $(2,6,3)$ |
| $(1,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,5,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,4,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,4)(2,6,3)$ | $-1$ |
| $3$ | $6$ | $(1,6,4,3,5,2)$ | $0$ |
| $3$ | $6$ | $(1,2,5,3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
