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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 2 + 16 a\cdot 31 + \left(5 a + 6\right)\cdot 31^{2} + \left(14 a + 10\right)\cdot 31^{3} + \left(6 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 19\cdot 31 + 29\cdot 31^{2} + 14\cdot 31^{3} + 3\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + \left(2 a + 23\right)\cdot 31 + 5 a\cdot 31^{2} + \left(22 a + 12\right)\cdot 31^{3} + \left(29 a + 13\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 10 + \left(28 a + 22\right)\cdot 31 + \left(25 a + 8\right)\cdot 31^{2} + \left(8 a + 20\right)\cdot 31^{3} + \left(a + 19\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 + 16\cdot 31 + 16\cdot 31^{2} + 2\cdot 31^{3} + 21\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 15 + \left(14 a + 11\right)\cdot 31 + 25 a\cdot 31^{2} + \left(16 a + 2\right)\cdot 31^{3} + \left(24 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $15$ |
$2$ |
$(1,2)$ |
$-1$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
