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:
| $r_{ 1 }$ |
$=$ |
$ 14 a + 61 + \left(62 a + 33\right)\cdot 67 + \left(41 a + 49\right)\cdot 67^{2} + \left(2 a + 24\right)\cdot 67^{3} + \left(32 a + 32\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 a + 27 + \left(30 a + 29\right)\cdot 67 + \left(63 a + 22\right)\cdot 67^{2} + \left(8 a + 45\right)\cdot 67^{3} + \left(25 a + 18\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 a + 50 + 4 a\cdot 67 + \left(25 a + 21\right)\cdot 67^{2} + \left(64 a + 60\right)\cdot 67^{3} + \left(34 a + 23\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 32\cdot 67 + 63\cdot 67^{2} + 48\cdot 67^{3} + 10\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 9 + \left(36 a + 55\right)\cdot 67 + \left(3 a + 44\right)\cdot 67^{2} + \left(58 a + 17\right)\cdot 67^{3} + \left(41 a + 43\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 49\cdot 67 + 66\cdot 67^{2} + 3\cdot 67^{3} + 5\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(1,2)(3,5)(4,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $6$ | $2$ | $(1,3)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $4$ | $3$ | $(1,3,4)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $1$ |
| $18$ | $4$ | $(1,5,3,2)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $1$ |
| $12$ | $6$ | $(1,3)(2,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
