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} + 8 a + 7 + \left(10 a^{2} + 2 a + 6\right)\cdot 13 + \left(10 a^{2} + 11 a\right)\cdot 13^{2} + \left(12 a^{2} + 11 a + 3\right)\cdot 13^{3} + \left(12 a^{2} + 12 a + 8\right)\cdot 13^{4} + \left(9 a^{2} + 12 a + 12\right)\cdot 13^{5} + \left(8 a^{2} + 3 a + 7\right)\cdot 13^{6} + \left(6 a^{2} + 10 a + 5\right)\cdot 13^{7} + \left(5 a^{2} + a\right)\cdot 13^{8} + \left(a^{2} + 8 a + 9\right)\cdot 13^{9} + \left(5 a^{2} + 11 a + 7\right)\cdot 13^{10} + \left(12 a^{2} + 9 a + 3\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a^{2} + 7 a + 6 + \left(9 a^{2} + 5 a\right)\cdot 13 + 8 a\cdot 13^{2} + \left(12 a^{2} + a + 2\right)\cdot 13^{3} + \left(8 a^{2} + 9 a + 7\right)\cdot 13^{4} + \left(12 a^{2} + 2 a + 7\right)\cdot 13^{5} + \left(8 a^{2} + 9 a + 12\right)\cdot 13^{6} + \left(3 a^{2} + a + 5\right)\cdot 13^{7} + \left(6 a^{2} + 12 a + 1\right)\cdot 13^{8} + \left(12 a^{2} + 10 a + 2\right)\cdot 13^{9} + \left(3 a^{2} + 8 a + 6\right)\cdot 13^{10} + \left(9 a^{2} + 6 a + 12\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a^{2} + 7 a + 4 + \left(9 a^{2} + 5 a + 9\right)\cdot 13 + \left(8 a + 1\right)\cdot 13^{2} + \left(12 a^{2} + a + 10\right)\cdot 13^{3} + \left(8 a^{2} + 9 a + 10\right)\cdot 13^{4} + \left(12 a^{2} + 2 a + 6\right)\cdot 13^{5} + \left(8 a^{2} + 9 a + 3\right)\cdot 13^{6} + \left(3 a^{2} + a + 7\right)\cdot 13^{7} + \left(6 a^{2} + 12 a + 6\right)\cdot 13^{8} + \left(12 a^{2} + 10 a + 5\right)\cdot 13^{9} + \left(3 a^{2} + 8 a + 5\right)\cdot 13^{10} + \left(9 a^{2} + 6 a\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a^{2} + 11 a + 8 + \left(6 a^{2} + 4 a + 5\right)\cdot 13 + \left(a^{2} + 6 a + 11\right)\cdot 13^{2} + \left(a^{2} + 12 a + 12\right)\cdot 13^{3} + \left(4 a^{2} + 3 a + 12\right)\cdot 13^{4} + \left(3 a^{2} + 10 a + 2\right)\cdot 13^{5} + \left(8 a^{2} + 12 a + 11\right)\cdot 13^{6} + \left(2 a^{2} + 5\right)\cdot 13^{7} + \left(a^{2} + 12 a + 8\right)\cdot 13^{8} + \left(12 a^{2} + 6 a\right)\cdot 13^{9} + \left(3 a^{2} + 5 a + 1\right)\cdot 13^{10} + \left(4 a^{2} + 9 a + 11\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a^{2} + 8 a + 5 + \left(10 a^{2} + 2 a + 2\right)\cdot 13 + \left(10 a^{2} + 11 a + 2\right)\cdot 13^{2} + \left(12 a^{2} + 11 a + 11\right)\cdot 13^{3} + \left(12 a^{2} + 12 a + 11\right)\cdot 13^{4} + \left(9 a^{2} + 12 a + 11\right)\cdot 13^{5} + \left(8 a^{2} + 3 a + 11\right)\cdot 13^{6} + \left(6 a^{2} + 10 a + 6\right)\cdot 13^{7} + \left(5 a^{2} + a + 5\right)\cdot 13^{8} + \left(a^{2} + 8 a + 12\right)\cdot 13^{9} + \left(5 a^{2} + 11 a + 6\right)\cdot 13^{10} + \left(12 a^{2} + 9 a + 4\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a^{2} + 8 a + 12 + \left(10 a^{2} + 2 a + 6\right)\cdot 13 + \left(10 a^{2} + 11 a + 1\right)\cdot 13^{2} + \left(12 a^{2} + 11 a + 11\right)\cdot 13^{3} + \left(12 a^{2} + 12 a + 5\right)\cdot 13^{4} + \left(9 a^{2} + 12 a + 2\right)\cdot 13^{5} + \left(8 a^{2} + 3 a + 2\right)\cdot 13^{6} + \left(6 a^{2} + 10 a + 1\right)\cdot 13^{7} + \left(5 a^{2} + a + 3\right)\cdot 13^{8} + \left(a^{2} + 8 a + 10\right)\cdot 13^{9} + \left(5 a^{2} + 11 a + 5\right)\cdot 13^{10} + \left(12 a^{2} + 9 a + 2\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 2 a^{2} + 7 a + 11 + \left(9 a^{2} + 5 a\right)\cdot 13 + \left(8 a + 1\right)\cdot 13^{2} + \left(12 a^{2} + a + 10\right)\cdot 13^{3} + \left(8 a^{2} + 9 a + 4\right)\cdot 13^{4} + \left(12 a^{2} + 2 a + 10\right)\cdot 13^{5} + \left(8 a^{2} + 9 a + 6\right)\cdot 13^{6} + \left(3 a^{2} + a + 1\right)\cdot 13^{7} + \left(6 a^{2} + 12 a + 4\right)\cdot 13^{8} + \left(12 a^{2} + 10 a + 3\right)\cdot 13^{9} + \left(3 a^{2} + 8 a + 4\right)\cdot 13^{10} + \left(9 a^{2} + 6 a + 11\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 5 a^{2} + 11 a + 2 + \left(6 a^{2} + 4 a + 10\right)\cdot 13 + \left(a^{2} + 6 a + 10\right)\cdot 13^{2} + \left(a^{2} + 12 a + 12\right)\cdot 13^{3} + \left(4 a^{2} + 3 a + 6\right)\cdot 13^{4} + \left(3 a^{2} + 10 a + 6\right)\cdot 13^{5} + \left(8 a^{2} + 12 a + 1\right)\cdot 13^{6} + 2 a^{2}13^{7} + \left(a^{2} + 12 a + 6\right)\cdot 13^{8} + \left(12 a^{2} + 6 a + 11\right)\cdot 13^{9} + \left(3 a^{2} + 5 a + 12\right)\cdot 13^{10} + \left(4 a^{2} + 9 a + 8\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 5 a^{2} + 11 a + 10 + \left(6 a^{2} + 4 a + 9\right)\cdot 13 + \left(a^{2} + 6 a + 9\right)\cdot 13^{2} + \left(a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(4 a^{2} + 3 a + 9\right)\cdot 13^{4} + \left(3 a^{2} + 10 a + 3\right)\cdot 13^{5} + \left(8 a^{2} + 12 a + 7\right)\cdot 13^{6} + \left(2 a^{2} + 4\right)\cdot 13^{7} + \left(a^{2} + 12 a + 3\right)\cdot 13^{8} + \left(12 a^{2} + 6 a + 10\right)\cdot 13^{9} + \left(3 a^{2} + 5 a + 1\right)\cdot 13^{10} + \left(4 a^{2} + 9 a + 10\right)\cdot 13^{11} +O\left(13^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,6,8,4,3,2)(5,7,9)$ |
| $(1,7)(3,9)(5,8)$ |
| $(2,5)(4,7)(6,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $3$ |
$2$ |
$(1,4)(2,8)(3,6)$ |
$0$ |
$0$ |
| $1$ |
$3$ |
$(1,8,3)(2,6,4)(5,9,7)$ |
$2 \zeta_{3}$ |
$-2 \zeta_{3} - 2$ |
| $1$ |
$3$ |
$(1,3,8)(2,4,6)(5,7,9)$ |
$-2 \zeta_{3} - 2$ |
$2 \zeta_{3}$ |
| $2$ |
$3$ |
$(1,9,2)(3,5,4)(6,8,7)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $2$ |
$3$ |
$(1,2,9)(3,4,5)(6,7,8)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $2$ |
$3$ |
$(1,4,7)(2,5,8)(3,6,9)$ |
$-1$ |
$-1$ |
| $3$ |
$6$ |
$(1,6,8,4,3,2)(5,7,9)$ |
$0$ |
$0$ |
| $3$ |
$6$ |
$(1,2,3,4,8,6)(5,9,7)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
