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