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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 4\cdot 31 + 15\cdot 31^{3} + 13\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 18 + \left(26 a + 20\right)\cdot 31 + \left(30 a + 10\right)\cdot 31^{2} + \left(a + 9\right)\cdot 31^{3} + \left(3 a + 24\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 13 + \left(15 a + 4\right)\cdot 31 + \left(21 a + 19\right)\cdot 31^{2} + \left(21 a + 18\right)\cdot 31^{3} + \left(28 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 + 15\cdot 31 + 31^{2} + 27\cdot 31^{3} + 15\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 22 + \left(15 a + 15\right)\cdot 31 + \left(9 a + 15\right)\cdot 31^{2} + \left(9 a + 9\right)\cdot 31^{3} + \left(2 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 5 + \left(4 a + 2\right)\cdot 31 + 15\cdot 31^{2} + \left(29 a + 13\right)\cdot 31^{3} + \left(27 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,5,4)$ |
| $(1,3,6,4,2,5)$ |
| $(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $3$ | $2$ | $(2,6)(3,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,6)(3,5,4)$ | $-1$ |
| $2$ | $6$ | $(1,3,6,4,2,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
