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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 24\cdot 29 + 20\cdot 29^{2} + 2\cdot 29^{3} + 23\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 28 + \left(4 a + 25\right)\cdot 29 + \left(16 a + 25\right)\cdot 29^{2} + \left(14 a + 3\right)\cdot 29^{3} + 17 a\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 a + 16 + \left(20 a + 7\right)\cdot 29 + \left(8 a + 7\right)\cdot 29^{2} + \left(3 a + 9\right)\cdot 29^{3} + \left(26 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 3\cdot 29 + 17\cdot 29^{2} + 22\cdot 29^{3} + 13\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a + 11 + \left(8 a + 26\right)\cdot 29 + 20 a\cdot 29^{2} + \left(25 a + 17\right)\cdot 29^{3} + \left(2 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a + 17 + \left(24 a + 28\right)\cdot 29 + \left(12 a + 14\right)\cdot 29^{2} + \left(14 a + 2\right)\cdot 29^{3} + \left(11 a + 15\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,3,5)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,4,5,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
