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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 9\cdot 29 + 20\cdot 29^{2} + 20\cdot 29^{3} + 7\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 22\cdot 29 + 10\cdot 29^{2} + 18\cdot 29^{3} + 18\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 15 + \left(4 a + 1\right)\cdot 29 + \left(20 a + 4\right)\cdot 29^{2} + \left(7 a + 18\right)\cdot 29^{3} + \left(22 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 1 + 23\cdot 29 + \left(20 a + 12\right)\cdot 29^{2} + \left(17 a + 20\right)\cdot 29^{3} + \left(28 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 3 + \left(24 a + 9\right)\cdot 29 + \left(8 a + 13\right)\cdot 29^{2} + \left(21 a + 7\right)\cdot 29^{3} + \left(6 a + 25\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 6 + \left(28 a + 22\right)\cdot 29 + \left(8 a + 25\right)\cdot 29^{2} + \left(11 a + 1\right)\cdot 29^{3} + 18\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,6)(4,5)$ |
| $(3,5)(4,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,3)$ | $0$ |
| $2$ | $3$ | $(1,6,4)(2,3,5)$ | $-1$ |
| $2$ | $6$ | $(1,5,6,2,4,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
