Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{3} + 6 x + 65 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 a^{2} + 18 a + 31 + \left(36 a^{2} + 56 a + 56\right)\cdot 67 + \left(19 a^{2} + 42 a + 55\right)\cdot 67^{2} + \left(17 a^{2} + 57 a + 46\right)\cdot 67^{3} + \left(62 a^{2} + 20 a + 25\right)\cdot 67^{4} + \left(34 a^{2} + 17 a + 50\right)\cdot 67^{5} + \left(49 a^{2} + 48 a + 41\right)\cdot 67^{6} + \left(56 a^{2} + 35 a + 3\right)\cdot 67^{7} + \left(64 a^{2} + 30 a + 36\right)\cdot 67^{8} + \left(37 a^{2} + 26 a + 62\right)\cdot 67^{9} + \left(61 a^{2} + 55 a + 22\right)\cdot 67^{10} + \left(62 a^{2} + 63 a + 28\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a^{2} + 53 a + 57 + \left(8 a^{2} + 9 a + 9\right)\cdot 67 + \left(40 a^{2} + 12 a + 4\right)\cdot 67^{2} + \left(47 a^{2} + 34\right)\cdot 67^{3} + \left(41 a^{2} + 54 a + 10\right)\cdot 67^{4} + \left(53 a^{2} + 6 a + 58\right)\cdot 67^{5} + \left(14 a^{2} + 45 a + 36\right)\cdot 67^{6} + \left(39 a^{2} + 17 a\right)\cdot 67^{7} + \left(43 a^{2} + 14 a + 18\right)\cdot 67^{8} + \left(13 a^{2} + 20 a + 32\right)\cdot 67^{9} + \left(3 a^{2} + 34 a + 57\right)\cdot 67^{10} + \left(27 a^{2} + 37 a + 18\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 + 18\cdot 67 + 41\cdot 67^{2} + 58\cdot 67^{3} + 31\cdot 67^{4} + 40\cdot 67^{5} + 36\cdot 67^{6} + 62\cdot 67^{7} + 14\cdot 67^{8} + 14\cdot 67^{9} + 41\cdot 67^{10} + 8\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 a^{2} + 8 a + 20 + \left(44 a^{2} + a + 22\right)\cdot 67 + \left(52 a^{2} + 38 a + 54\right)\cdot 67^{2} + \left(45 a^{2} + 6 a + 26\right)\cdot 67^{3} + \left(28 a^{2} + 6 a + 25\right)\cdot 67^{4} + \left(40 a + 46\right)\cdot 67^{5} + \left(21 a^{2} + 5 a + 61\right)\cdot 67^{6} + \left(64 a^{2} + 65 a + 33\right)\cdot 67^{7} + \left(60 a^{2} + 18 a + 20\right)\cdot 67^{8} + \left(58 a^{2} + 25 a + 12\right)\cdot 67^{9} + \left(23 a^{2} + 11 a + 6\right)\cdot 67^{10} + \left(32 a^{2} + 38 a + 40\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 60 a^{2} + 41 a + 17 + \left(52 a^{2} + 9 a + 55\right)\cdot 67 + \left(61 a^{2} + 53 a + 23\right)\cdot 67^{2} + \left(3 a^{2} + 2 a + 60\right)\cdot 67^{3} + \left(43 a^{2} + 40 a + 15\right)\cdot 67^{4} + \left(31 a^{2} + 9 a + 37\right)\cdot 67^{5} + \left(63 a^{2} + 13 a + 30\right)\cdot 67^{6} + \left(12 a^{2} + 33 a + 29\right)\cdot 67^{7} + \left(8 a^{2} + 17 a + 10\right)\cdot 67^{8} + \left(37 a^{2} + 15 a + 59\right)\cdot 67^{9} + \left(48 a^{2} + 37\right)\cdot 67^{10} + \left(38 a^{2} + 32 a + 65\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 24\cdot 67 + 16\cdot 67^{2} + 16\cdot 67^{3} + 12\cdot 67^{4} + 39\cdot 67^{5} + 10\cdot 67^{6} + 53\cdot 67^{7} + 49\cdot 67^{8} + 25\cdot 67^{9} + 33\cdot 67^{10} + 43\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 a^{2} + 32 a + 48 + \left(23 a^{2} + 54 a + 5\right)\cdot 67 + \left(18 a^{2} + 33 a + 51\right)\cdot 67^{2} + \left(52 a^{2} + 17 a + 52\right)\cdot 67^{3} + \left(45 a^{2} + 15 a + 26\right)\cdot 67^{4} + \left(59 a^{2} + 14 a + 15\right)\cdot 67^{5} + \left(30 a^{2} + 14 a + 34\right)\cdot 67^{6} + \left(21 a^{2} + 35 a + 63\right)\cdot 67^{7} + \left(35 a^{2} + 61 a + 51\right)\cdot 67^{8} + \left(3 a^{2} + 33 a + 58\right)\cdot 67^{9} + \left(27 a^{2} + 4 a + 18\right)\cdot 67^{10} + \left(62 a^{2} + 15 a + 26\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 13 a^{2} + 49 a + 30 + \left(35 a^{2} + 2 a + 51\right)\cdot 67 + \left(8 a^{2} + 21 a + 11\right)\cdot 67^{2} + \left(34 a^{2} + 49 a + 47\right)\cdot 67^{3} + \left(46 a^{2} + 64 a + 29\right)\cdot 67^{4} + \left(20 a^{2} + 45 a + 60\right)\cdot 67^{5} + \left(21 a^{2} + 7 a + 62\right)\cdot 67^{6} + \left(6 a^{2} + 14 a + 2\right)\cdot 67^{7} + \left(55 a^{2} + 58 a + 64\right)\cdot 67^{8} + \left(49 a^{2} + 12 a + 42\right)\cdot 67^{9} + \left(36 a^{2} + 28 a + 57\right)\cdot 67^{10} + \left(44 a^{2} + 14 a + 21\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 21 + 23\cdot 67 + 9\cdot 67^{2} + 59\cdot 67^{3} + 22\cdot 67^{4} + 54\cdot 67^{5} + 19\cdot 67^{6} + 18\cdot 67^{7} + 2\cdot 67^{8} + 27\cdot 67^{9} + 59\cdot 67^{10} + 14\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5,4)(2,7,8)(3,6,9)$ |
| $(2,8,7)(3,6,9)$ |
| $(1,2,9)(3,5,7)(4,8,6)$ |
| $(2,3)(6,7)(8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,6)(3,4)(5,9)$ | $1$ |
| $1$ | $3$ | $(1,5,4)(2,7,8)(3,6,9)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,4,5)(2,8,7)(3,9,6)$ | $-3 \zeta_{3} - 3$ |
| $6$ | $3$ | $(1,2,9)(3,5,7)(4,8,6)$ | $0$ |
| $6$ | $3$ | $(1,8,9)(2,3,5)(4,7,6)$ | $0$ |
| $6$ | $3$ | $(2,8,7)(3,6,9)$ | $0$ |
| $6$ | $3$ | $(1,9,7)(2,4,6)(3,8,5)$ | $0$ |
| $9$ | $6$ | $(1,3,5,6,4,9)(2,8,7)$ | $-\zeta_{3} - 1$ |
| $9$ | $6$ | $(1,9,4,6,5,3)(2,7,8)$ | $\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
