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