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