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:
| $r_{ 1 }$ |
$=$ |
$ 52 a + 9 + \left(56 a + 25\right)\cdot 67 + \left(63 a + 30\right)\cdot 67^{2} + \left(54 a + 29\right)\cdot 67^{3} + \left(31 a + 28\right)\cdot 67^{4} + \left(35 a + 25\right)\cdot 67^{5} + \left(44 a + 59\right)\cdot 67^{6} + \left(4 a + 33\right)\cdot 67^{7} + \left(33 a + 23\right)\cdot 67^{8} + \left(58 a + 58\right)\cdot 67^{9} + \left(15 a + 48\right)\cdot 67^{10} + \left(62 a + 46\right)\cdot 67^{11} + \left(7 a + 30\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 61 + 31\cdot 67 + 23\cdot 67^{2} + 20\cdot 67^{3} + 26\cdot 67^{4} + 23\cdot 67^{5} + 40\cdot 67^{6} + 48\cdot 67^{7} + 31\cdot 67^{8} + 36\cdot 67^{9} + 48\cdot 67^{10} + 37\cdot 67^{11} + 28\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 36 + \left(29 a + 42\right)\cdot 67 + \left(52 a + 28\right)\cdot 67^{2} + \left(14 a + 29\right)\cdot 67^{3} + \left(33 a + 2\right)\cdot 67^{4} + \left(58 a + 5\right)\cdot 67^{5} + \left(24 a + 27\right)\cdot 67^{6} + \left(a + 59\right)\cdot 67^{7} + \left(52 a + 8\right)\cdot 67^{8} + \left(10 a + 46\right)\cdot 67^{9} + \left(18 a + 34\right)\cdot 67^{10} + \left(35 a + 29\right)\cdot 67^{11} + \left(65 a + 53\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 a + 41 + \left(37 a + 7\right)\cdot 67 + \left(14 a + 8\right)\cdot 67^{2} + \left(52 a + 36\right)\cdot 67^{3} + \left(33 a + 53\right)\cdot 67^{4} + \left(8 a + 4\right)\cdot 67^{5} + \left(42 a + 1\right)\cdot 67^{6} + \left(65 a + 40\right)\cdot 67^{7} + \left(14 a + 14\right)\cdot 67^{8} + \left(56 a + 37\right)\cdot 67^{9} + \left(48 a + 29\right)\cdot 67^{10} + \left(31 a + 18\right)\cdot 67^{11} + \left(a + 12\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + 16 + \left(10 a + 66\right)\cdot 67 + \left(3 a + 27\right)\cdot 67^{2} + \left(12 a + 51\right)\cdot 67^{3} + \left(35 a + 33\right)\cdot 67^{4} + \left(31 a + 1\right)\cdot 67^{5} + \left(22 a + 1\right)\cdot 67^{6} + \left(62 a + 8\right)\cdot 67^{7} + \left(33 a + 17\right)\cdot 67^{8} + \left(8 a + 58\right)\cdot 67^{9} + \left(51 a + 53\right)\cdot 67^{10} + \left(4 a + 11\right)\cdot 67^{11} + 59 a\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 + 27\cdot 67 + 15\cdot 67^{2} + 34\cdot 67^{3} + 56\cdot 67^{4} + 6\cdot 67^{5} + 5\cdot 67^{6} + 11\cdot 67^{7} + 38\cdot 67^{8} + 31\cdot 67^{9} + 52\cdot 67^{10} + 56\cdot 67^{11} + 8\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,2,3)(5,6)$ |
| $(1,2,5)$ |
| $(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $4$ | $3$ | $(3,4,6)$ | $-2$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $9$ | $4$ | $(1,4,2,3)(5,6)$ | $0$ |
| $9$ | $4$ | $(1,3,2,4)(5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
