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:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 17 + \left(24 a + 8\right)\cdot 31 + \left(11 a + 28\right)\cdot 31^{2} + \left(8 a + 6\right)\cdot 31^{3} + \left(a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 19 + \left(29 a + 4\right)\cdot 31 + \left(28 a + 19\right)\cdot 31^{2} + \left(8 a + 10\right)\cdot 31^{3} + \left(20 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 8\cdot 31 + 19\cdot 31^{2} + 22\cdot 31^{3} + 4\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 25 + \left(a + 28\right)\cdot 31 + \left(2 a + 16\right)\cdot 31^{2} + \left(22 a + 30\right)\cdot 31^{3} + \left(10 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 5\cdot 31 + 13\cdot 31^{2} + 10\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 28 + \left(6 a + 5\right)\cdot 31 + \left(19 a + 27\right)\cdot 31^{2} + \left(22 a + 11\right)\cdot 31^{3} + \left(29 a + 15\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,3,5,4,2)$ |
| $(1,6)(2,5)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ |
| $15$ | $2$ | $(1,6)(3,5)$ | $1$ |
| $20$ | $3$ | $(1,3,4)(2,6,5)$ | $-1$ |
| $30$ | $4$ | $(1,5,6,3)$ | $1$ |
| $24$ | $5$ | $(2,3,5,4,6)$ | $0$ |
| $20$ | $6$ | $(1,6,3,5,4,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
