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