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