Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 9 + \left(5 a + 27\right)\cdot 31 + \left(29 a + 19\right)\cdot 31^{2} + \left(10 a + 26\right)\cdot 31^{3} + \left(25 a + 27\right)\cdot 31^{4} + \left(a + 27\right)\cdot 31^{5} + \left(5 a + 24\right)\cdot 31^{6} + \left(13 a + 28\right)\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 5 + 9 a\cdot 31 + \left(18 a + 15\right)\cdot 31^{2} + \left(15 a + 6\right)\cdot 31^{3} + 5\cdot 31^{4} + \left(21 a + 15\right)\cdot 31^{5} + \left(14 a + 18\right)\cdot 31^{6} + \left(9 a + 5\right)\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 21\cdot 31 + 4\cdot 31^{2} + 21\cdot 31^{3} + 31^{4} + 18\cdot 31^{5} + 20\cdot 31^{6} + 29\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a + 9 + \left(21 a + 16\right)\cdot 31 + \left(12 a + 11\right)\cdot 31^{2} + \left(15 a + 19\right)\cdot 31^{3} + \left(30 a + 21\right)\cdot 31^{4} + \left(9 a + 25\right)\cdot 31^{5} + \left(16 a + 26\right)\cdot 31^{6} + \left(21 a + 9\right)\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 29 + \left(25 a + 27\right)\cdot 31 + \left(a + 10\right)\cdot 31^{2} + \left(20 a + 19\right)\cdot 31^{3} + \left(5 a + 5\right)\cdot 31^{4} + \left(29 a + 6\right)\cdot 31^{5} + \left(25 a + 2\right)\cdot 31^{6} + \left(17 a + 19\right)\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,5,4,3,2)$ |
| $(1,3,4,2)$ |
| $(1,4)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $5$ |
$2$ |
$(1,2)(3,5)$ |
$0$ |
| $5$ |
$4$ |
$(1,5,2,3)$ |
$0$ |
| $5$ |
$4$ |
$(1,3,2,5)$ |
$0$ |
| $4$ |
$5$ |
$(1,5,4,3,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
