Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{3} + 2 x + 11 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a^{2} + 7 a + 6 + \left(5 a^{2} + 8 a + 11\right)\cdot 13 + \left(12 a^{2} + a + 10\right)\cdot 13^{2} + \left(12 a^{2} + 5 a + 1\right)\cdot 13^{3} + \left(3 a + 6\right)\cdot 13^{4} + \left(5 a^{2} + 7 a + 1\right)\cdot 13^{5} + \left(11 a^{2} + 8 a + 1\right)\cdot 13^{6} + \left(7 a^{2} + 10 a + 11\right)\cdot 13^{7} + \left(10 a^{2} + 9 a + 7\right)\cdot 13^{8} + \left(9 a^{2} + 7 a + 6\right)\cdot 13^{9} + \left(9 a^{2} + 2 a + 4\right)\cdot 13^{10} + \left(a^{2} + 7\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 3\cdot 13 + 7\cdot 13^{2} + 8\cdot 13^{3} + 6\cdot 13^{4} + 13^{5} + 8\cdot 13^{6} + 13^{7} + 5\cdot 13^{8} + 2\cdot 13^{9} + 8\cdot 13^{10} + 3\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a^{2} + 4 a + 10 + \left(6 a^{2} + 11 a + 9\right)\cdot 13 + \left(3 a^{2} + 11 a + 7\right)\cdot 13^{2} + \left(12 a^{2} + 10 a + 7\right)\cdot 13^{3} + \left(4 a^{2} + 6\right)\cdot 13^{4} + \left(12 a^{2} + 2 a + 1\right)\cdot 13^{5} + \left(4 a^{2} + 4 a + 4\right)\cdot 13^{6} + \left(5 a^{2} + 6 a + 8\right)\cdot 13^{7} + \left(12 a^{2} + 12 a + 10\right)\cdot 13^{8} + \left(7 a^{2} + 6\right)\cdot 13^{9} + \left(5 a^{2} + a\right)\cdot 13^{10} + \left(12 a^{2} + 4 a + 7\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a^{2} + 2 a + 3 + \left(4 a + 9\right)\cdot 13 + \left(a^{2} + 2 a + 8\right)\cdot 13^{2} + \left(6 a^{2} + a + 5\right)\cdot 13^{3} + \left(2 a^{2} + a + 12\right)\cdot 13^{4} + \left(9 a^{2} + 11 a + 6\right)\cdot 13^{5} + \left(11 a^{2} + 8 a + 1\right)\cdot 13^{6} + \left(11 a^{2} + 3 a + 12\right)\cdot 13^{7} + \left(8 a^{2} + 10 a + 9\right)\cdot 13^{8} + \left(5 a^{2} + a + 9\right)\cdot 13^{9} + \left(6 a^{2} + 5 a + 8\right)\cdot 13^{10} + \left(6 a^{2} + 12 a\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 11 + \left(7 a^{2} + 4\right)\cdot 13 + \left(12 a^{2} + 9 a + 2\right)\cdot 13^{2} + \left(6 a^{2} + 6 a + 11\right)\cdot 13^{3} + \left(9 a^{2} + 8 a + 8\right)\cdot 13^{4} + \left(11 a^{2} + 7 a + 1\right)\cdot 13^{5} + \left(2 a^{2} + 8 a + 7\right)\cdot 13^{6} + \left(6 a^{2} + 11 a + 4\right)\cdot 13^{7} + \left(6 a^{2} + 5 a + 2\right)\cdot 13^{8} + \left(10 a^{2} + 3 a + 3\right)\cdot 13^{9} + \left(9 a^{2} + 5 a\right)\cdot 13^{10} + \left(4 a^{2} + 7\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 + 7\cdot 13 + 12\cdot 13^{3} + 4\cdot 13^{4} + 7\cdot 13^{5} + 2\cdot 13^{6} + 6\cdot 13^{7} + 5\cdot 13^{8} + 2\cdot 13^{9} + 12\cdot 13^{10} + 9\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 4 a^{2} + 3 a + 12 + \left(4 a^{2} + 10 a + 10\right)\cdot 13 + \left(12 a^{2} + 10\right)\cdot 13^{2} + \left(4 a^{2} + 8 a + 10\right)\cdot 13^{3} + \left(6 a^{2} + 8 a + 12\right)\cdot 13^{4} + \left(9 a^{2} + 8 a + 1\right)\cdot 13^{5} + \left(10 a^{2} + 12 a + 3\right)\cdot 13^{6} + \left(8 a^{2} + 7 a + 4\right)\cdot 13^{7} + \left(5 a^{2} + 9 a + 10\right)\cdot 13^{8} + \left(9 a^{2} + 7 a + 8\right)\cdot 13^{9} + \left(3 a^{2} + 10 a + 6\right)\cdot 13^{10} + \left(7 a^{2} + 9 a + 4\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 6 a + 11 + \left(2 a^{2} + 4 a + 7\right)\cdot 13 + \left(10 a^{2} + 3\right)\cdot 13^{2} + \left(8 a^{2} + 7 a + 7\right)\cdot 13^{3} + \left(a^{2} + 3 a + 6\right)\cdot 13^{4} + \left(4 a^{2} + 2 a + 3\right)\cdot 13^{5} + \left(10 a^{2} + 9 a + 11\right)\cdot 13^{6} + \left(11 a^{2} + 11 a + 3\right)\cdot 13^{7} + \left(7 a^{2} + 3 a\right)\cdot 13^{8} + \left(8 a^{2} + 4 a + 12\right)\cdot 13^{9} + \left(3 a^{2} + a + 10\right)\cdot 13^{10} + \left(6 a^{2} + 12 a + 11\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,7,3)(2,8,6,4)$ |
| $(2,3,4)(5,8,6)$ |
| $(1,8,7,4)(2,3,6,5)$ |
| $(1,7)(2,6)(3,5)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,6)(3,5)(4,8)$ |
$-2$ |
$-2$ |
| $4$ |
$3$ |
$(1,8,2)(4,6,7)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $4$ |
$3$ |
$(1,2,8)(4,7,6)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $6$ |
$4$ |
$(1,8,7,4)(2,3,6,5)$ |
$0$ |
$0$ |
| $4$ |
$6$ |
$(1,6,8,7,2,4)(3,5)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $4$ |
$6$ |
$(1,4,2,7,8,6)(3,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
