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:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 17 + \left(9 a + 23\right)\cdot 31 + \left(6 a + 28\right)\cdot 31^{2} + \left(8 a + 16\right)\cdot 31^{3} + \left(29 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 18 + \left(9 a + 9\right)\cdot 31 + \left(29 a + 5\right)\cdot 31^{2} + \left(a + 18\right)\cdot 31^{3} + 16 a\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 13\cdot 31 + 20\cdot 31^{2} + 17\cdot 31^{3} + 10\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 4\cdot 31 + 14\cdot 31^{2} + 20\cdot 31^{3} + 29\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 5 + \left(21 a + 18\right)\cdot 31 + 24 a\cdot 31^{2} + \left(22 a + 27\right)\cdot 31^{3} + \left(a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 28 + \left(21 a + 22\right)\cdot 31 + \left(a + 23\right)\cdot 31^{2} + \left(29 a + 23\right)\cdot 31^{3} + \left(14 a + 30\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(1,3)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $1$ |
| $4$ | $3$ | $(1,3,5)$ | $-2$ |
| $18$ | $4$ | $(1,4,3,2)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $-1$ |
| $12$ | $6$ | $(1,3)(2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
