Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 54 a + 60 + \left(47 a + 31\right)\cdot 67 + \left(45 a + 66\right)\cdot 67^{2} + \left(18 a + 18\right)\cdot 67^{3} + \left(64 a + 48\right)\cdot 67^{4} + \left(36 a + 58\right)\cdot 67^{5} + \left(30 a + 57\right)\cdot 67^{6} + \left(35 a + 44\right)\cdot 67^{7} + \left(16 a + 51\right)\cdot 67^{8} + \left(48 a + 45\right)\cdot 67^{9} + \left(47 a + 62\right)\cdot 67^{10} + \left(17 a + 21\right)\cdot 67^{11} + \left(59 a + 24\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 + 22\cdot 67 + 61\cdot 67^{2} + 26\cdot 67^{3} + 50\cdot 67^{4} + 6\cdot 67^{5} + 26\cdot 67^{6} + 38\cdot 67^{7} + 65\cdot 67^{8} + 20\cdot 67^{9} + 49\cdot 67^{10} + 54\cdot 67^{11} + 47\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 8 + \left(19 a + 35\right)\cdot 67 + 21 a\cdot 67^{2} + \left(48 a + 48\right)\cdot 67^{3} + \left(2 a + 18\right)\cdot 67^{4} + \left(30 a + 8\right)\cdot 67^{5} + \left(36 a + 9\right)\cdot 67^{6} + \left(31 a + 22\right)\cdot 67^{7} + \left(50 a + 15\right)\cdot 67^{8} + \left(18 a + 21\right)\cdot 67^{9} + \left(19 a + 4\right)\cdot 67^{10} + \left(49 a + 45\right)\cdot 67^{11} + \left(7 a + 42\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 a + 11 + \left(66 a + 23\right)\cdot 67 + \left(16 a + 66\right)\cdot 67^{2} + \left(45 a + 51\right)\cdot 67^{3} + \left(32 a + 57\right)\cdot 67^{4} + \left(47 a + 21\right)\cdot 67^{5} + 22\cdot 67^{6} + \left(31 a + 5\right)\cdot 67^{7} + \left(33 a + 49\right)\cdot 67^{8} + \left(34 a + 14\right)\cdot 67^{9} + \left(3 a + 10\right)\cdot 67^{10} + \left(29 a + 44\right)\cdot 67^{11} + \left(12 a + 56\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 57 + 43\cdot 67 + 50 a\cdot 67^{2} + \left(21 a + 15\right)\cdot 67^{3} + \left(34 a + 9\right)\cdot 67^{4} + \left(19 a + 45\right)\cdot 67^{5} + \left(66 a + 44\right)\cdot 67^{6} + \left(35 a + 61\right)\cdot 67^{7} + \left(33 a + 17\right)\cdot 67^{8} + \left(32 a + 52\right)\cdot 67^{9} + \left(63 a + 56\right)\cdot 67^{10} + \left(37 a + 22\right)\cdot 67^{11} + \left(54 a + 10\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 + 44\cdot 67 + 5\cdot 67^{2} + 40\cdot 67^{3} + 16\cdot 67^{4} + 60\cdot 67^{5} + 40\cdot 67^{6} + 28\cdot 67^{7} + 67^{8} + 46\cdot 67^{9} + 17\cdot 67^{10} + 12\cdot 67^{11} + 19\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,2)(3,5,6)$ |
| $(1,4)(3,5)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(4,5)$ | $-1$ |
| $6$ | $2$ | $(2,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,4,2)(3,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,3,4)$ | $-1$ |
| $6$ | $4$ | $(1,3)(2,4,6,5)$ | $1$ |
| $8$ | $6$ | $(1,5,6,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
