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:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 1 + \left(30 a + 1\right)\cdot 31 + \left(5 a + 26\right)\cdot 31^{2} + \left(8 a + 15\right)\cdot 31^{3} + \left(18 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 12 + \left(25 a + 6\right)\cdot 31 + \left(19 a + 7\right)\cdot 31^{2} + \left(24 a + 26\right)\cdot 31^{3} + \left(5 a + 27\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 22 + 5\cdot 31 + \left(25 a + 7\right)\cdot 31^{2} + \left(22 a + 26\right)\cdot 31^{3} + \left(12 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 28 + \left(5 a + 17\right)\cdot 31 + \left(11 a + 21\right)\cdot 31^{2} + \left(6 a + 24\right)\cdot 31^{3} + \left(25 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $6$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
