Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 55 a + 39 + \left(6 a + 5\right)\cdot 67 + \left(26 a + 10\right)\cdot 67^{2} + \left(23 a + 63\right)\cdot 67^{3} + \left(62 a + 50\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 29 + \left(60 a + 2\right)\cdot 67 + \left(40 a + 62\right)\cdot 67^{2} + \left(43 a + 54\right)\cdot 67^{3} + \left(4 a + 57\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 9 + \left(60 a + 45\right)\cdot 67 + \left(40 a + 43\right)\cdot 67^{2} + \left(43 a + 49\right)\cdot 67^{3} + \left(4 a + 37\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 a + 10 + \left(6 a + 30\right)\cdot 67 + \left(26 a + 31\right)\cdot 67^{2} + \left(23 a + 54\right)\cdot 67^{3} + \left(62 a + 32\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 a + 57 + \left(6 a + 5\right)\cdot 67 + \left(26 a + 13\right)\cdot 67^{2} + \left(23 a + 49\right)\cdot 67^{3} + \left(62 a + 12\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a + 58 + \left(60 a + 44\right)\cdot 67 + \left(40 a + 40\right)\cdot 67^{2} + \left(43 a + 63\right)\cdot 67^{3} + \left(4 a + 8\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,5,6,4,3)$ |
| $(1,6)(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,5,4)(2,6,3)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,4,5)(2,3,6)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,5,6,4,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,4,6,5,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
