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:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 25 + \left(18 a + 20\right)\cdot 29 + \left(5 a + 9\right)\cdot 29^{2} + \left(20 a + 10\right)\cdot 29^{3} + 8\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 13 + \left(10 a + 11\right)\cdot 29 + \left(21 a + 24\right)\cdot 29^{2} + \left(21 a + 28\right)\cdot 29^{3} + \left(5 a + 10\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 26\cdot 29 + 21\cdot 29^{2} + 24\cdot 29^{3} + 3\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 4 + \left(10 a + 8\right)\cdot 29 + \left(23 a + 19\right)\cdot 29^{2} + \left(8 a + 18\right)\cdot 29^{3} + \left(28 a + 20\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 16 + \left(18 a + 17\right)\cdot 29 + \left(7 a + 4\right)\cdot 29^{2} + 7 a\cdot 29^{3} + \left(23 a + 18\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 + 2\cdot 29 + 7\cdot 29^{2} + 4\cdot 29^{3} + 25\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6)(2,3,4)$ |
| $(2,5)(3,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
| $8$ | $3$ | $(1,5,6)(2,3,4)$ | $0$ |
| $6$ | $4$ | $(1,5,4,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
