Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 17\cdot 29 + 10\cdot 29^{2} + 16\cdot 29^{3} + 3\cdot 29^{4} + 20\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 15 + 18\cdot 29 + \left(7 a + 6\right)\cdot 29^{2} + 28\cdot 29^{3} + \left(12 a + 2\right)\cdot 29^{4} + \left(14 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 23 + \left(28 a + 2\right)\cdot 29 + \left(21 a + 12\right)\cdot 29^{2} + \left(28 a + 22\right)\cdot 29^{3} + \left(16 a + 4\right)\cdot 29^{4} + \left(14 a + 3\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 15\cdot 29 + 28\cdot 29^{2} + 4\cdot 29^{3} + 18\cdot 29^{4} + 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 6 + \left(24 a + 20\right)\cdot 29 + \left(11 a + 11\right)\cdot 29^{2} + \left(2 a + 7\right)\cdot 29^{3} + \left(16 a + 4\right)\cdot 29^{4} + \left(20 a + 16\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 a + 18 + \left(4 a + 12\right)\cdot 29 + \left(17 a + 17\right)\cdot 29^{2} + \left(26 a + 7\right)\cdot 29^{3} + \left(12 a + 24\right)\cdot 29^{4} + \left(8 a + 15\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(5,6)$ |
| $(2,3)(4,5)$ |
| $(4,5,6)$ |
| $(1,4)(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,4)(3,5)$ | $0$ |
| $9$ | $2$ | $(1,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $2$ | $3$ | $(1,2,3)(4,6,5)$ | $-2$ |
| $4$ | $3$ | $(4,5,6)$ | $1$ |
| $6$ | $6$ | $(1,5,2,6,3,4)$ | $0$ |
| $6$ | $6$ | $(1,6,2,5,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
