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