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:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 10 + \left(14 a + 13\right)\cdot 31 + \left(20 a + 8\right)\cdot 31^{2} + \left(30 a + 6\right)\cdot 31^{3} + \left(28 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 17 + \left(16 a + 23\right)\cdot 31 + \left(10 a + 3\right)\cdot 31^{2} + 16\cdot 31^{3} + \left(2 a + 29\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 a + 20 + 23\cdot 31 + \left(25 a + 5\right)\cdot 31^{2} + \left(12 a + 2\right)\cdot 31^{3} + \left(30 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 14 + \left(30 a + 28\right)\cdot 31 + \left(5 a + 23\right)\cdot 31^{2} + \left(18 a + 2\right)\cdot 31^{3} + 3\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 + 4\cdot 31 + 20\cdot 31^{2} + 3\cdot 31^{3} + 10\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
