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:
| $r_{ 1 }$ |
$=$ |
$ 9 + 16\cdot 31 + 12\cdot 31^{2} + 18\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a + 18 + \left(15 a + 3\right)\cdot 31 + \left(12 a + 20\right)\cdot 31^{2} + \left(12 a + 15\right)\cdot 31^{3} + \left(16 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 4 + \left(15 a + 11\right)\cdot 31 + \left(18 a + 29\right)\cdot 31^{2} + \left(18 a + 27\right)\cdot 31^{3} + \left(14 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 24 + 17 a\cdot 31 + \left(25 a + 22\right)\cdot 31^{2} + \left(23 a + 16\right)\cdot 31^{3} + \left(25 a + 20\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 14\cdot 31 + 15\cdot 31^{2} + 6\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 2 + \left(13 a + 16\right)\cdot 31 + \left(5 a + 24\right)\cdot 31^{2} + \left(7 a + 7\right)\cdot 31^{3} + \left(5 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2)$ |
| $(1,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $6$ | $2$ | $(1,2)$ | $-2$ |
| $9$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)$ | $1$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $0$ |
| $12$ | $6$ | $(1,2)(4,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
