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:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 24 + \left(29 a + 30\right)\cdot 31 + \left(25 a + 20\right)\cdot 31^{2} + \left(17 a + 25\right)\cdot 31^{3} + 2 a\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 + 26\cdot 31 + 28\cdot 31^{2} + 11\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 13 + \left(10 a + 4\right)\cdot 31 + \left(5 a + 12\right)\cdot 31^{2} + \left(30 a + 22\right)\cdot 31^{3} + \left(25 a + 29\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 10 + \left(a + 4\right)\cdot 31 + \left(5 a + 12\right)\cdot 31^{2} + \left(13 a + 4\right)\cdot 31^{3} + \left(28 a + 19\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 + 27\cdot 31 + 6\cdot 31^{2} + 24\cdot 31^{3} + 11\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 5 + \left(20 a + 30\right)\cdot 31 + \left(25 a + 11\right)\cdot 31^{2} + 15\cdot 31^{3} + \left(5 a + 20\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(3,5,6)$ |
| $(1,6)(2,5)(3,4)$ |
| $(1,4,2)$ |
| $(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,4,2)(3,5,6)$ | $-2$ |
| $2$ | $3$ | $(1,4,2)(3,6,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,2)$ | $1$ |
| $6$ | $6$ | $(1,3,4,5,2,6)$ | $0$ |
| $6$ | $6$ | $(1,5,4,3,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
