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:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 1 + \left(5 a + 2\right)\cdot 31 + \left(30 a + 10\right)\cdot 31^{2} + \left(4 a + 19\right)\cdot 31^{3} + \left(28 a + 7\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 5 + \left(25 a + 3\right)\cdot 31 + 27\cdot 31^{2} + \left(26 a + 10\right)\cdot 31^{3} + \left(2 a + 20\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 3 + \left(25 a + 4\right)\cdot 31 + 21\cdot 31^{2} + \left(26 a + 21\right)\cdot 31^{3} + \left(2 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 19 + \left(5 a + 15\right)\cdot 31 + \left(30 a + 3\right)\cdot 31^{2} + 4 a\cdot 31^{3} + 28 a\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 18 + \left(25 a + 20\right)\cdot 31 + 2\cdot 31^{2} + \left(26 a + 30\right)\cdot 31^{3} + \left(2 a + 27\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 17 + \left(5 a + 16\right)\cdot 31 + \left(30 a + 28\right)\cdot 31^{2} + \left(4 a + 10\right)\cdot 31^{3} + \left(28 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4,5,6,2)$ |
| $(1,5)(2,4)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,4,6)(2,3,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,4)(2,5,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,4,5,6,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,6,5,4,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
