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:
| $r_{ 1 }$ |
$=$ |
$ 9 + 12\cdot 31 + 16\cdot 31^{2} + 21\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 3 + \left(5 a + 6\right)\cdot 31 + \left(4 a + 2\right)\cdot 31^{2} + \left(15 a + 14\right)\cdot 31^{3} + \left(29 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 9 + \left(12 a + 16\right)\cdot 31 + \left(27 a + 13\right)\cdot 31^{2} + \left(3 a + 18\right)\cdot 31^{3} + \left(9 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 7 + \left(18 a + 12\right)\cdot 31 + \left(3 a + 24\right)\cdot 31^{2} + \left(27 a + 29\right)\cdot 31^{3} + \left(21 a + 22\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 5 + \left(25 a + 15\right)\cdot 31 + \left(26 a + 5\right)\cdot 31^{2} + \left(15 a + 9\right)\cdot 31^{3} + \left(a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
