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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 22\cdot 29 + 17\cdot 29^{2} + 4\cdot 29^{3} + 15\cdot 29^{4} + 10\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 11 + \left(9 a + 6\right)\cdot 29 + \left(16 a + 25\right)\cdot 29^{2} + \left(5 a + 6\right)\cdot 29^{3} + \left(4 a + 6\right)\cdot 29^{4} + \left(20 a + 18\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 14\cdot 29 + 10\cdot 29^{2} + 5\cdot 29^{3} + 2\cdot 29^{4} + 12\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 9 + \left(19 a + 8\right)\cdot 29 + \left(12 a + 10\right)\cdot 29^{2} + \left(23 a + 18\right)\cdot 29^{3} + \left(24 a + 21\right)\cdot 29^{4} + \left(8 a + 27\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 1 + \left(25 a + 10\right)\cdot 29 + \left(10 a + 26\right)\cdot 29^{2} + \left(5 a + 17\right)\cdot 29^{3} + \left(6 a + 22\right)\cdot 29^{4} + \left(2 a + 6\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 20 + \left(3 a + 25\right)\cdot 29 + \left(18 a + 25\right)\cdot 29^{2} + \left(23 a + 4\right)\cdot 29^{3} + \left(22 a + 19\right)\cdot 29^{4} + \left(26 a + 11\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,6)$ |
| $(3,6,5)$ |
| $(1,2)(5,6)$ |
| $(1,6)(2,3)(4,5)$ |
| $(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $9$ | $2$ | $(1,4)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,4,2)(3,5,6)$ | $-2$ |
| $2$ | $3$ | $(1,4,2)(3,6,5)$ | $-2$ |
| $4$ | $3$ | $(1,2,4)$ | $1$ |
| $6$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
| $6$ | $6$ | $(1,5,2,6,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
