Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 24 + \left(3 a + 7\right)\cdot 31 + \left(23 a + 10\right)\cdot 31^{2} + \left(8 a + 29\right)\cdot 31^{3} + \left(a + 28\right)\cdot 31^{4} + \left(30 a + 30\right)\cdot 31^{5} + \left(15 a + 23\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 9\cdot 31 + 11\cdot 31^{2} + 2\cdot 31^{3} + 21\cdot 31^{4} + 13\cdot 31^{5} + 26\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 26 + \left(25 a + 24\right)\cdot 31 + \left(15 a + 5\right)\cdot 31^{2} + \left(28 a + 29\right)\cdot 31^{3} + 7\cdot 31^{4} + \left(12 a + 14\right)\cdot 31^{5} + \left(22 a + 7\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 3 + \left(27 a + 9\right)\cdot 31 + \left(7 a + 22\right)\cdot 31^{2} + \left(22 a + 23\right)\cdot 31^{3} + \left(29 a + 22\right)\cdot 31^{4} + 27\cdot 31^{5} + \left(15 a + 25\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 30 + \left(5 a + 10\right)\cdot 31 + \left(15 a + 12\right)\cdot 31^{2} + \left(2 a + 8\right)\cdot 31^{3} + \left(30 a + 12\right)\cdot 31^{4} + \left(18 a + 6\right)\cdot 31^{5} + \left(8 a + 9\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,5,2,4)$ |
| $(1,2)(4,5)$ |
| $(1,2,4,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $5$ |
$2$ |
$(1,2)(4,5)$ |
$0$ |
| $5$ |
$4$ |
$(1,5,2,4)$ |
$0$ |
| $5$ |
$4$ |
$(1,4,2,5)$ |
$0$ |
| $4$ |
$5$ |
$(1,2,4,3,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
