Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 20 + \left(13 a + 11\right)\cdot 31 + \left(26 a + 11\right)\cdot 31^{2} + \left(2 a + 15\right)\cdot 31^{3} + \left(a + 18\right)\cdot 31^{4} + \left(25 a + 10\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 25\cdot 31 + 25\cdot 31^{2} + 21\cdot 31^{3} + 24\cdot 31^{4} + 30\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 3 + 29 a\cdot 31 + \left(11 a + 15\right)\cdot 31^{2} + \left(7 a + 9\right)\cdot 31^{3} + \left(13 a + 20\right)\cdot 31^{4} + \left(16 a + 22\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 19 + \left(a + 19\right)\cdot 31 + \left(19 a + 9\right)\cdot 31^{2} + \left(23 a + 12\right)\cdot 31^{3} + \left(17 a + 8\right)\cdot 31^{4} + \left(14 a + 11\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 + 10\cdot 31 + 11\cdot 31^{2} + 8\cdot 31^{3} + 3\cdot 31^{4} + 20\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 a + 15 + \left(17 a + 25\right)\cdot 31 + \left(4 a + 19\right)\cdot 31^{2} + \left(28 a + 25\right)\cdot 31^{3} + \left(29 a + 17\right)\cdot 31^{4} + \left(5 a + 28\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(3,4)$ |
| $(2,5)(3,4)$ |
| $(1,2,4)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,4)(3,6,5)$ | $0$ |
| $4$ | $3$ | $(1,4,2)(3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
