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 + 10\cdot 31 + 15\cdot 31^{2} + 30\cdot 31^{3} + 9\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 23 + \left(17 a + 25\right)\cdot 31 + 15 a\cdot 31^{2} + \left(29 a + 18\right)\cdot 31^{3} + 15 a\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 4 + \left(13 a + 23\right)\cdot 31 + \left(15 a + 14\right)\cdot 31^{2} + \left(a + 30\right)\cdot 31^{3} + \left(15 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 17 + \left(8 a + 21\right)\cdot 31 + \left(26 a + 24\right)\cdot 31^{2} + \left(27 a + 7\right)\cdot 31^{3} + \left(23 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 9 + \left(22 a + 12\right)\cdot 31 + \left(4 a + 6\right)\cdot 31^{2} + \left(3 a + 6\right)\cdot 31^{3} + \left(7 a + 3\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
