Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 16\cdot 31 + 8\cdot 31^{2} + 3\cdot 31^{3} + 9\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 8\cdot 31 + 25\cdot 31^{2} + 30\cdot 31^{3} + 27\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 20 + \left(29 a + 4\right)\cdot 31 + \left(a + 24\right)\cdot 31^{2} + \left(6 a + 8\right)\cdot 31^{3} + \left(25 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 4 + \left(a + 10\right)\cdot 31 + \left(29 a + 29\right)\cdot 31^{2} + \left(24 a + 18\right)\cdot 31^{3} + \left(5 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 24 + \left(11 a + 19\right)\cdot 31 + \left(30 a + 24\right)\cdot 31^{2} + \left(13 a + 16\right)\cdot 31^{3} + \left(9 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 11 + \left(19 a + 2\right)\cdot 31 + 12\cdot 31^{2} + \left(17 a + 14\right)\cdot 31^{3} + \left(21 a + 19\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,5)$ |
| $(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(1,3)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $1$ |
| $4$ | $3$ | $(1,3,4)$ | $-2$ |
| $18$ | $4$ | $(1,5,3,2)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $-1$ |
| $12$ | $6$ | $(1,3)(2,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
