Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 61 a + 46 + \left(65 a + 32\right)\cdot 67 + \left(48 a + 35\right)\cdot 67^{2} + \left(51 a + 21\right)\cdot 67^{3} + \left(53 a + 52\right)\cdot 67^{4} + \left(61 a + 3\right)\cdot 67^{5} + \left(60 a + 43\right)\cdot 67^{6} + \left(50 a + 62\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 22 + \left(a + 34\right)\cdot 67 + \left(18 a + 31\right)\cdot 67^{2} + \left(15 a + 45\right)\cdot 67^{3} + \left(13 a + 14\right)\cdot 67^{4} + \left(5 a + 63\right)\cdot 67^{5} + \left(6 a + 23\right)\cdot 67^{6} + \left(16 a + 4\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 1 + 32\cdot 67 + \left(36 a + 10\right)\cdot 67^{2} + \left(52 a + 24\right)\cdot 67^{3} + \left(18 a + 43\right)\cdot 67^{4} + \left(27 a + 10\right)\cdot 67^{5} + \left(a + 47\right)\cdot 67^{6} + \left(5 a + 3\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 + 55\cdot 67 + 8\cdot 67^{2} + 28\cdot 67^{3} + 67^{4} + 21\cdot 67^{5} + 13\cdot 67^{6} + 46\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 a + \left(66 a + 35\right)\cdot 67 + \left(30 a + 56\right)\cdot 67^{2} + \left(14 a + 42\right)\cdot 67^{3} + \left(48 a + 23\right)\cdot 67^{4} + \left(39 a + 56\right)\cdot 67^{5} + \left(65 a + 19\right)\cdot 67^{6} + \left(61 a + 63\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 63 + 35\cdot 67 + \left(36 a + 46\right)\cdot 67^{2} + \left(52 a + 2\right)\cdot 67^{3} + \left(18 a + 1\right)\cdot 67^{4} + \left(27 a + 33\right)\cdot 67^{5} + \left(a + 41\right)\cdot 67^{6} + \left(5 a + 44\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 12 + 11\cdot 67 + 58\cdot 67^{2} + 38\cdot 67^{3} + 65\cdot 67^{4} + 45\cdot 67^{5} + 53\cdot 67^{6} + 20\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 a + 5 + \left(66 a + 31\right)\cdot 67 + \left(30 a + 20\right)\cdot 67^{2} + \left(14 a + 64\right)\cdot 67^{3} + \left(48 a + 65\right)\cdot 67^{4} + \left(39 a + 33\right)\cdot 67^{5} + \left(65 a + 25\right)\cdot 67^{6} + \left(61 a + 22\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,6)(2,4,8)$ |
| $(1,8,2,6)(3,7,5,4)$ |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,3,2,5)(4,6,7,8)$ |
| $(1,4)(2,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $-2$ |
| $12$ | $2$ | $(1,4)(2,7)(6,8)$ | $0$ |
| $8$ | $3$ | $(3,7,8)(4,6,5)$ | $-1$ |
| $6$ | $4$ | $(1,8,2,6)(3,7,5,4)$ | $0$ |
| $8$ | $6$ | $(1,2)(3,6,7,5,8,4)$ | $1$ |
| $6$ | $8$ | $(1,3,7,6,2,5,4,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,5,7,8,2,3,4,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
