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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 23\cdot 31 + 8\cdot 31^{2} + 2\cdot 31^{3} + 11\cdot 31^{4} +O\left(31^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 9 + \left(7 a + 8\right)\cdot 31 + \left(28 a + 9\right)\cdot 31^{2} + 22\cdot 31^{3} + \left(13 a + 8\right)\cdot 31^{4} + \left(8 a + 9\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 20 + \left(23 a + 2\right)\cdot 31 + \left(2 a + 27\right)\cdot 31^{2} + \left(30 a + 26\right)\cdot 31^{3} + \left(17 a + 2\right)\cdot 31^{4} + \left(22 a + 13\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 1 + \left(19 a + 3\right)\cdot 31 + \left(16 a + 17\right)\cdot 31^{2} + \left(22 a + 6\right)\cdot 31^{3} + \left(10 a + 20\right)\cdot 31^{4} + \left(9 a + 15\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 6 + \left(11 a + 24\right)\cdot 31 + \left(14 a + 30\right)\cdot 31^{2} + \left(8 a + 3\right)\cdot 31^{3} + \left(20 a + 19\right)\cdot 31^{4} + \left(21 a + 23\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(2,4,3,5)$ |
| $(2,3)(4,5)$ |
| $(1,3,4,5,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $5$ |
$2$ |
$(2,3)(4,5)$ |
$0$ |
| $5$ |
$4$ |
$(2,4,3,5)$ |
$0$ |
| $5$ |
$4$ |
$(2,5,3,4)$ |
$0$ |
| $4$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
