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 + 27 + \left(7 a + 17\right)\cdot 29 + \left(3 a + 14\right)\cdot 29^{2} + \left(11 a + 3\right)\cdot 29^{3} + \left(11 a + 4\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 24 + \left(7 a + 20\right)\cdot 29 + \left(3 a + 5\right)\cdot 29^{2} + \left(11 a + 2\right)\cdot 29^{3} + \left(11 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 6 + \left(21 a + 8\right)\cdot 29 + \left(25 a + 23\right)\cdot 29^{2} + \left(17 a + 26\right)\cdot 29^{3} + \left(17 a + 20\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 12\cdot 29 + 20\cdot 29^{2} + 7\cdot 29^{3} + 15\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 3 + \left(21 a + 11\right)\cdot 29 + \left(25 a + 14\right)\cdot 29^{2} + \left(17 a + 25\right)\cdot 29^{3} + \left(17 a + 24\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 16\cdot 29 + 8\cdot 29^{2} + 21\cdot 29^{3} + 13\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,6)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,3,4)(2,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,5,4,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
