Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13\cdot 29 + 5\cdot 29^{2} + 27\cdot 29^{3} + 25\cdot 29^{4} + 7\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 1 + 16\cdot 29 + 23\cdot 29^{2} + 29^{3} + 3\cdot 29^{4} + 21\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 5 + 21\cdot 29 + \left(25 a + 10\right)\cdot 29^{2} + 27 a\cdot 29^{3} + \left(7 a + 14\right)\cdot 29^{4} + \left(27 a + 11\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 11 + \left(25 a + 10\right)\cdot 29^{2} + \left(27 a + 1\right)\cdot 29^{3} + \left(7 a + 3\right)\cdot 29^{4} + \left(27 a + 5\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 19 + \left(28 a + 28\right)\cdot 29 + \left(3 a + 18\right)\cdot 29^{2} + \left(a + 27\right)\cdot 29^{3} + \left(21 a + 25\right)\cdot 29^{4} + \left(a + 23\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 25 + \left(28 a + 7\right)\cdot 29 + \left(3 a + 18\right)\cdot 29^{2} + \left(a + 28\right)\cdot 29^{3} + \left(21 a + 14\right)\cdot 29^{4} + \left(a + 17\right)\cdot 29^{5} +O\left(29^{ 6 }\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.
