Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 8\cdot 31 + 20\cdot 31^{2} + 27\cdot 31^{3} + 2\cdot 31^{4} + 26\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 7 + \left(6 a + 23\right)\cdot 31 + \left(15 a + 7\right)\cdot 31^{2} + \left(10 a + 26\right)\cdot 31^{3} + 23 a\cdot 31^{4} + \left(14 a + 25\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 7\cdot 31 + 29\cdot 31^{2} + 14\cdot 31^{3} + 20\cdot 31^{4} + 9\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 10 + \left(8 a + 7\right)\cdot 31 + \left(a + 5\right)\cdot 31^{2} + \left(17 a + 26\right)\cdot 31^{3} + \left(6 a + 17\right)\cdot 31^{4} + \left(18 a + 16\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 a + 14 + \left(24 a + 17\right)\cdot 31 + 15 a\cdot 31^{2} + \left(20 a + 1\right)\cdot 31^{3} + \left(7 a + 6\right)\cdot 31^{4} + 16 a\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 2 + \left(22 a + 29\right)\cdot 31 + \left(29 a + 29\right)\cdot 31^{2} + \left(13 a + 27\right)\cdot 31^{3} + \left(24 a + 13\right)\cdot 31^{4} + \left(12 a + 15\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$8$ |
$8$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
$0$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
