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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 9 + \left(8 a + 7\right)\cdot 29 + \left(15 a + 23\right)\cdot 29^{2} + \left(11 a + 26\right)\cdot 29^{3} + \left(24 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a + 20 + \left(27 a + 6\right)\cdot 29 + \left(27 a + 21\right)\cdot 29^{2} + \left(10 a + 3\right)\cdot 29^{3} + \left(25 a + 28\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 22 + \left(20 a + 1\right)\cdot 29 + \left(13 a + 4\right)\cdot 29^{2} + \left(17 a + 11\right)\cdot 29^{3} + \left(4 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 5\cdot 29 + 6\cdot 29^{2} + 2\cdot 29^{3} + 24\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 29 + 15\cdot 29^{2} + 12\cdot 29^{3} + 6\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 24 + \left(a + 5\right)\cdot 29 + \left(a + 17\right)\cdot 29^{2} + \left(18 a + 1\right)\cdot 29^{3} + \left(3 a + 28\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6)(2,3,4)$ |
| $(1,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $15$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $20$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
| $12$ | $5$ | $(1,6,2,3,4)$ | $-1$ |
| $12$ | $5$ | $(1,2,4,6,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
