Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 4\cdot 29 + 6\cdot 29^{2} + 12\cdot 29^{3} + 27\cdot 29^{4} + 8\cdot 29^{5} + 7\cdot 29^{6} + 17\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 5 + \left(20 a + 5\right)\cdot 29 + \left(13 a + 27\right)\cdot 29^{2} + \left(16 a + 28\right)\cdot 29^{3} + \left(14 a + 4\right)\cdot 29^{4} + \left(10 a + 18\right)\cdot 29^{5} + \left(18 a + 6\right)\cdot 29^{6} + \left(3 a + 5\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 16 + \left(8 a + 11\right)\cdot 29 + \left(15 a + 17\right)\cdot 29^{2} + \left(12 a + 10\right)\cdot 29^{3} + \left(14 a + 3\right)\cdot 29^{4} + \left(18 a + 27\right)\cdot 29^{5} + 10 a\cdot 29^{6} + \left(25 a + 5\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 8 + \left(a + 10\right)\cdot 29 + \left(18 a + 17\right)\cdot 29^{2} + \left(8 a + 19\right)\cdot 29^{3} + \left(13 a + 25\right)\cdot 29^{4} + \left(13 a + 3\right)\cdot 29^{5} + \left(24 a + 25\right)\cdot 29^{6} + \left(23 a + 25\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 26 + \left(27 a + 26\right)\cdot 29 + \left(10 a + 18\right)\cdot 29^{2} + \left(20 a + 15\right)\cdot 29^{3} + \left(15 a + 25\right)\cdot 29^{4} + \left(15 a + 28\right)\cdot 29^{5} + \left(4 a + 17\right)\cdot 29^{6} + \left(5 a + 4\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,5)(3,4)$ |
| $(1,4,5,3)$ |
| $(1,4,2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $5$ | $2$ | $(1,5)(3,4)$ | $0$ |
| $5$ | $4$ | $(1,4,5,3)$ | $0$ |
| $5$ | $4$ | $(1,3,5,4)$ | $0$ |
| $4$ | $5$ | $(1,4,2,3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
