Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 21 + \left(28 a + 19\right)\cdot 29 + \left(13 a + 18\right)\cdot 29^{2} + \left(7 a + 3\right)\cdot 29^{3} + \left(7 a + 20\right)\cdot 29^{4} + \left(9 a + 18\right)\cdot 29^{5} + \left(2 a + 23\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 5 + 27\cdot 29 + \left(15 a + 1\right)\cdot 29^{2} + \left(21 a + 27\right)\cdot 29^{3} + \left(21 a + 19\right)\cdot 29^{4} + \left(19 a + 28\right)\cdot 29^{5} + \left(26 a + 25\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 28\cdot 29 + 25\cdot 29^{2} + 19\cdot 29^{3} + 4\cdot 29^{4} + 8\cdot 29^{5} + 21\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + \left(28 a + 18\right)\cdot 29 + \left(13 a + 16\right)\cdot 29^{2} + \left(7 a + 8\right)\cdot 29^{3} + \left(7 a + 19\right)\cdot 29^{4} + \left(9 a + 10\right)\cdot 29^{5} + \left(2 a + 5\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 13 + 25\cdot 29 + \left(15 a + 28\right)\cdot 29^{2} + \left(21 a + 2\right)\cdot 29^{3} + \left(21 a + 19\right)\cdot 29^{4} + \left(19 a + 20\right)\cdot 29^{5} + \left(26 a + 7\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 + 26\cdot 29 + 23\cdot 29^{2} + 24\cdot 29^{3} + 3\cdot 29^{4} + 3\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,4)(2,5)(3,6)$ |
| $(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,2,4,3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
