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:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 33 + 58 a\cdot 67 + \left(3 a + 58\right)\cdot 67^{2} + \left(52 a + 13\right)\cdot 67^{3} + 4 a\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 53 + \left(58 a + 49\right)\cdot 67 + \left(3 a + 10\right)\cdot 67^{2} + \left(52 a + 37\right)\cdot 67^{3} + \left(4 a + 33\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 a + 50 + \left(8 a + 11\right)\cdot 67 + \left(63 a + 15\right)\cdot 67^{2} + \left(14 a + 17\right)\cdot 67^{3} + \left(62 a + 34\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 a + 3 + \left(8 a + 61\right)\cdot 67 + \left(63 a + 34\right)\cdot 67^{2} + \left(14 a + 40\right)\cdot 67^{3} + 62 a\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 23 + \left(58 a + 33\right)\cdot 67 + \left(3 a + 62\right)\cdot 67^{2} + \left(52 a + 10\right)\cdot 67^{3} + \left(4 a + 49\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 a + 40 + \left(8 a + 44\right)\cdot 67 + \left(63 a + 19\right)\cdot 67^{2} + \left(14 a + 14\right)\cdot 67^{3} + \left(62 a + 16\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2,5)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,2)(3,6,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,4,5,3,2,6)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,6,2,3,5,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
