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