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