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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 8\cdot 31 + 24\cdot 31^{2} + 9\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 14 + \left(20 a + 8\right)\cdot 31 + \left(11 a + 17\right)\cdot 31^{2} + 20 a\cdot 31^{3} + \left(18 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 22 + \left(10 a + 13\right)\cdot 31 + \left(19 a + 20\right)\cdot 31^{2} + \left(10 a + 29\right)\cdot 31^{3} + \left(12 a + 3\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 30 + \left(a + 23\right)\cdot 31 + \left(18 a + 9\right)\cdot 31^{2} + \left(7 a + 30\right)\cdot 31^{3} + 19 a\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 + 19\cdot 31 + 7\cdot 31^{2} + 4\cdot 31^{3} + 29\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 15 + \left(29 a + 18\right)\cdot 31 + \left(12 a + 13\right)\cdot 31^{2} + \left(23 a + 27\right)\cdot 31^{3} + 11 a\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2)$ |
| $(1,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $6$ | $2$ | $(1,2)$ | $0$ |
| $9$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $4$ | $3$ | $(4,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ |
| $12$ | $6$ | $(1,4,2,5,3,6)$ | $1$ |
| $12$ | $6$ | $(1,2)(4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
