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:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 1 + \left(3 a + 6\right)\cdot 29 + \left(8 a + 7\right)\cdot 29^{2} + \left(2 a + 5\right)\cdot 29^{3} + \left(27 a + 14\right)\cdot 29^{4} + 23\cdot 29^{5} + \left(6 a + 12\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 9 + \left(20 a + 28\right)\cdot 29 + \left(11 a + 28\right)\cdot 29^{2} + \left(13 a + 12\right)\cdot 29^{3} + \left(2 a + 10\right)\cdot 29^{4} + \left(19 a + 22\right)\cdot 29^{5} + \left(11 a + 1\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 11 + \left(25 a + 19\right)\cdot 29 + \left(20 a + 15\right)\cdot 29^{2} + \left(26 a + 8\right)\cdot 29^{3} + \left(a + 2\right)\cdot 29^{4} + \left(28 a + 1\right)\cdot 29^{5} + \left(22 a + 13\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 3\cdot 29 + 20\cdot 29^{2} + 18\cdot 29^{3} + 20\cdot 29^{4} + 23\cdot 29^{5} + 21\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 19 + \left(8 a + 10\right)\cdot 29 + \left(17 a + 9\right)\cdot 29^{2} + \left(15 a + 10\right)\cdot 29^{3} + \left(26 a + 9\right)\cdot 29^{4} + \left(9 a + 28\right)\cdot 29^{5} + \left(17 a + 11\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 + 19\cdot 29 + 5\cdot 29^{2} + 2\cdot 29^{3} + 29^{4} + 17\cdot 29^{5} + 25\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)(2,5,6)$ |
| $(3,4)(5,6)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $-1$ |
| $6$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,3,4)(2,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,2,3)$ | $-1$ |
| $6$ | $4$ | $(1,6,2,4)(3,5)$ | $1$ |
| $8$ | $6$ | $(1,5,6,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
