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:
| $r_{ 1 }$ |
$=$ |
$ 19 + 30\cdot 31 + 14\cdot 31^{2} + 13\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 26 + \left(5 a + 24\right)\cdot 31 + \left(24 a + 4\right)\cdot 31^{2} + \left(21 a + 21\right)\cdot 31^{3} + \left(a + 15\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 6 + \left(26 a + 1\right)\cdot 31 + \left(9 a + 7\right)\cdot 31^{2} + \left(21 a + 22\right)\cdot 31^{3} + \left(30 a + 27\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 3 + 25 a\cdot 31 + \left(6 a + 17\right)\cdot 31^{2} + \left(9 a + 9\right)\cdot 31^{3} + \left(29 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 5 + \left(4 a + 8\right)\cdot 31 + 21 a\cdot 31^{2} + \left(9 a + 24\right)\cdot 31^{3} + 5\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 + 28\cdot 31 + 17\cdot 31^{2} + 2\cdot 31^{3} + 30\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,2)(4,6,5)$ |
| $(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $20$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $12$ | $5$ | $(1,3,6,4,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,6,2,3,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
