Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9\cdot 13 + 7\cdot 13^{2} + 7\cdot 13^{3} + 2\cdot 13^{4} + 2\cdot 13^{5} + 2\cdot 13^{6} + 2\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 11 + \left(4 a + 5\right)\cdot 13 + \left(8 a + 6\right)\cdot 13^{2} + \left(4 a + 4\right)\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} + \left(11 a + 12\right)\cdot 13^{5} + \left(4 a + 12\right)\cdot 13^{6} + \left(12 a + 10\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 10 + \left(2 a + 3\right)\cdot 13 + \left(2 a + 3\right)\cdot 13^{2} + \left(2 a + 4\right)\cdot 13^{3} + 7\cdot 13^{4} + \left(8 a + 12\right)\cdot 13^{5} + \left(11 a + 12\right)\cdot 13^{6} + \left(7 a + 4\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 5 + \left(10 a + 11\right)\cdot 13 + \left(10 a + 2\right)\cdot 13^{2} + \left(10 a + 4\right)\cdot 13^{3} + \left(12 a + 5\right)\cdot 13^{4} + \left(4 a + 7\right)\cdot 13^{5} + \left(a + 3\right)\cdot 13^{6} + \left(5 a + 1\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 + 4\cdot 13 + 6\cdot 13^{2} + 5\cdot 13^{3} + 10\cdot 13^{4} + 2\cdot 13^{5} + 13^{6} + 7\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 9 + \left(6 a + 1\right)\cdot 13 + 4\cdot 13^{2} + \left(7 a + 9\right)\cdot 13^{3} + \left(3 a + 7\right)\cdot 13^{4} + 3 a\cdot 13^{5} + \left(9 a + 10\right)\cdot 13^{6} + \left(6 a + 4\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 3 a + 8 + 8 a\cdot 13 + \left(4 a + 10\right)\cdot 13^{2} + 8 a\cdot 13^{3} + \left(a + 4\right)\cdot 13^{4} + a\cdot 13^{5} + \left(8 a + 6\right)\cdot 13^{6} + 5\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 7 a + 2 + \left(6 a + 2\right)\cdot 13 + \left(12 a + 11\right)\cdot 13^{2} + \left(5 a + 2\right)\cdot 13^{3} + \left(9 a + 4\right)\cdot 13^{4} + 9 a\cdot 13^{5} + \left(3 a + 3\right)\cdot 13^{6} + \left(6 a + 2\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,5,6)(2,8,7,4)$ |
| $(1,3,4)(5,6,8)$ |
| $(1,6)(3,5)(4,8)$ |
| $(1,7,5,2)(3,8,6,4)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,6)(3,5)(4,8)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,3,4)(5,6,8)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,3,5,6)(2,8,7,4)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,5)(2,8,6,7,4,3)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,4,6,7,5,8,3,2)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,8,6,2,5,4,3,7)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
