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