Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{3} + 3 x + 86 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a^{2} + 82 a + 50 + \left(79 a^{2} + 69 a + 39\right)\cdot 89 + \left(6 a^{2} + 66 a + 43\right)\cdot 89^{2} + \left(70 a^{2} + 78 a + 21\right)\cdot 89^{3} + \left(87 a^{2} + 82 a + 27\right)\cdot 89^{4} + \left(a^{2} + 59 a + 63\right)\cdot 89^{5} + \left(4 a^{2} + 64 a + 37\right)\cdot 89^{6} + \left(4 a^{2} + 15 a + 67\right)\cdot 89^{7} + \left(82 a^{2} + 44 a + 15\right)\cdot 89^{8} + \left(66 a^{2} + 23 a + 15\right)\cdot 89^{9} +O\left(89^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a^{2} + 22 a + 46 + \left(87 a^{2} + 61 a + 55\right)\cdot 89 + \left(13 a^{2} + 27 a + 57\right)\cdot 89^{2} + \left(72 a^{2} + 16 a + 25\right)\cdot 89^{3} + \left(83 a^{2} + 77 a + 19\right)\cdot 89^{4} + \left(44 a^{2} + 71 a + 60\right)\cdot 89^{5} + \left(a^{2} + a + 32\right)\cdot 89^{6} + \left(36 a^{2} + 30 a + 42\right)\cdot 89^{7} + \left(5 a^{2} + 60 a + 40\right)\cdot 89^{8} + \left(73 a^{2} + 79 a + 27\right)\cdot 89^{9} +O\left(89^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 79 + 78\cdot 89 + 42\cdot 89^{2} + 55\cdot 89^{3} + 14\cdot 89^{4} + 7\cdot 89^{5} + 84\cdot 89^{7} + 7\cdot 89^{9} +O\left(89^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 85 a^{2} + 19 a + 22 + \left(49 a^{2} + 12 a + 70\right)\cdot 89 + \left(40 a^{2} + 30 a + 21\right)\cdot 89^{2} + \left(23 a^{2} + 73 a + 17\right)\cdot 89^{3} + \left(41 a^{2} + 47 a + 23\right)\cdot 89^{4} + \left(79 a^{2} + 46 a + 40\right)\cdot 89^{5} + \left(17 a^{2} + 45 a + 65\right)\cdot 89^{6} + \left(43 a^{2} + 36 a + 56\right)\cdot 89^{7} + \left(76 a^{2} + 54 a + 4\right)\cdot 89^{8} + \left(12 a^{2} + 83 a + 85\right)\cdot 89^{9} +O\left(89^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 + 67\cdot 89 + 8\cdot 89^{2} + 3\cdot 89^{3} + 15\cdot 89^{4} + 50\cdot 89^{5} + 21\cdot 89^{6} + 58\cdot 89^{7} + 76\cdot 89^{8} + 13\cdot 89^{9} +O\left(89^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 56 a^{2} + 63 a + 53 + \left(19 a^{2} + 10 a + 9\right)\cdot 89 + \left(85 a^{2} + 69 a + 22\right)\cdot 89^{2} + \left(76 a^{2} + 5 a + 35\right)\cdot 89^{3} + \left(53 a^{2} + 56 a + 48\right)\cdot 89^{4} + \left(39 a^{2} + 36 a + 49\right)\cdot 89^{5} + \left(49 a^{2} + 48 a + 39\right)\cdot 89^{6} + \left(12 a^{2} + 70 a + 84\right)\cdot 89^{7} + \left(24 a^{2} + 18 a + 77\right)\cdot 89^{8} + \left(11 a^{2} + 55 a + 81\right)\cdot 89^{9} +O\left(89^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 71 a^{2} + 74 a + 83 + \left(11 a^{2} + 46 a + 82\right)\cdot 89 + \left(68 a^{2} + 83 a + 76\right)\cdot 89^{2} + \left(35 a^{2} + 82 a + 41\right)\cdot 89^{3} + \left(6 a^{2} + 17 a + 42\right)\cdot 89^{4} + \left(42 a^{2} + 46 a + 54\right)\cdot 89^{5} + \left(83 a^{2} + 22 a + 18\right)\cdot 89^{6} + \left(48 a^{2} + 43 a + 68\right)\cdot 89^{7} + \left(a^{2} + 73 a + 32\right)\cdot 89^{8} + \left(38 a^{2} + 74 a + 46\right)\cdot 89^{9} +O\left(89^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 37 a^{2} + 7 a + 15 + \left(19 a^{2} + 66 a + 9\right)\cdot 89 + \left(52 a^{2} + 78 a + 45\right)\cdot 89^{2} + \left(77 a^{2} + 9 a + 36\right)\cdot 89^{3} + \left(82 a^{2} + 74 a + 17\right)\cdot 89^{4} + \left(58 a^{2} + 5 a + 88\right)\cdot 89^{5} + \left(21 a^{2} + 84 a + 72\right)\cdot 89^{6} + \left(33 a^{2} + 70 a + 36\right)\cdot 89^{7} + \left(77 a^{2} + 15 a + 6\right)\cdot 89^{8} + \left(64 a^{2} + 39 a + 11\right)\cdot 89^{9} +O\left(89^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 77 + 31\cdot 89 + 37\cdot 89^{2} + 30\cdot 89^{3} + 59\cdot 89^{4} + 31\cdot 89^{5} + 67\cdot 89^{6} + 35\cdot 89^{7} + 11\cdot 89^{8} + 68\cdot 89^{9} +O\left(89^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,7)(3,6)(4,9)(5,8)$ |
| $(1,8,9,7,4,5,2,6,3)$ |
| $(3,5,9)(4,6,8)$ |
| $(1,2,7)(3,9,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,9)(2,5)(3,7)(4,8)$ | $0$ |
| $2$ | $3$ | $(1,7,2)(3,9,5)(4,6,8)$ | $-3$ |
| $3$ | $3$ | $(1,2,7)(3,9,5)$ | $0$ |
| $3$ | $3$ | $(1,7,2)(3,5,9)$ | $0$ |
| $9$ | $6$ | $(1,5,7,9,2,3)(4,8)$ | $0$ |
| $9$ | $6$ | $(1,3,2,9,7,5)(4,8)$ | $0$ |
| $6$ | $9$ | $(1,8,9,7,4,5,2,6,3)$ | $0$ |
| $6$ | $9$ | $(1,9,4,2,3,8,7,5,6)$ | $0$ |
| $6$ | $9$ | $(1,6,9,7,8,5,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
