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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 7 + 12\cdot 29 + \left(11 a + 1\right)\cdot 29^{2} + \left(11 a + 20\right)\cdot 29^{3} + \left(15 a + 21\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 21 + 8\cdot 29 + \left(11 a + 15\right)\cdot 29^{2} + \left(11 a + 21\right)\cdot 29^{3} + \left(15 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 19 + \left(28 a + 2\right)\cdot 29 + \left(17 a + 14\right)\cdot 29^{2} + \left(17 a + 22\right)\cdot 29^{3} + \left(13 a + 5\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 1 + \left(28 a + 23\right)\cdot 29 + \left(17 a + 26\right)\cdot 29^{2} + \left(17 a + 7\right)\cdot 29^{3} + 13 a\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 15 + \left(28 a + 19\right)\cdot 29 + \left(17 a + 11\right)\cdot 29^{2} + \left(17 a + 9\right)\cdot 29^{3} + \left(13 a + 5\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 25 + 20\cdot 29 + \left(11 a + 17\right)\cdot 29^{2} + \left(11 a + 5\right)\cdot 29^{3} + \left(15 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6,4,2,3)$ |
| $(1,4)(2,5)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
