Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 a + 84 + \left(53 a + 40\right)\cdot 89 + \left(8 a + 10\right)\cdot 89^{2} + \left(24 a + 35\right)\cdot 89^{3} + \left(14 a + 7\right)\cdot 89^{4} + \left(11 a + 45\right)\cdot 89^{5} + \left(27 a + 79\right)\cdot 89^{6} + \left(11 a + 69\right)\cdot 89^{7} + \left(85 a + 34\right)\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 83 a + 50 + \left(78 a + 74\right)\cdot 89 + \left(7 a + 43\right)\cdot 89^{2} + \left(30 a + 30\right)\cdot 89^{3} + \left(9 a + 64\right)\cdot 89^{4} + \left(79 a + 3\right)\cdot 89^{5} + \left(30 a + 82\right)\cdot 89^{6} + \left(12 a + 46\right)\cdot 89^{7} + \left(88 a + 33\right)\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 a + 46 + \left(58 a + 2\right)\cdot 89 + \left(66 a + 31\right)\cdot 89^{2} + \left(41 a + 22\right)\cdot 89^{3} + \left(20 a + 45\right)\cdot 89^{4} + \left(48 a + 10\right)\cdot 89^{5} + 5\cdot 89^{6} + \left(71 a + 33\right)\cdot 89^{7} + \left(65 a + 3\right)\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 + 30\cdot 89 + 50\cdot 89^{2} + 76\cdot 89^{3} + 14\cdot 89^{4} + 73\cdot 89^{5} + 35\cdot 89^{6} + 19\cdot 89^{7} + 38\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 81\cdot 89 + 10\cdot 89^{2} + 49\cdot 89^{3} + 72\cdot 89^{4} + 39\cdot 89^{5} + 71\cdot 89^{6} + 55\cdot 89^{7} + 19\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 8 + \left(10 a + 10\right)\cdot 89 + \left(81 a + 20\right)\cdot 89^{2} + \left(58 a + 55\right)\cdot 89^{3} + \left(79 a + 10\right)\cdot 89^{4} + \left(9 a + 14\right)\cdot 89^{5} + \left(58 a + 41\right)\cdot 89^{6} + \left(76 a + 13\right)\cdot 89^{7} + 15\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 a + 75 + \left(30 a + 1\right)\cdot 89 + \left(22 a + 83\right)\cdot 89^{2} + \left(47 a + 69\right)\cdot 89^{3} + \left(68 a + 57\right)\cdot 89^{4} + \left(40 a + 60\right)\cdot 89^{5} + \left(88 a + 49\right)\cdot 89^{6} + \left(17 a + 84\right)\cdot 89^{7} + \left(23 a + 36\right)\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 56 a + 48 + \left(35 a + 25\right)\cdot 89 + \left(80 a + 17\right)\cdot 89^{2} + \left(64 a + 17\right)\cdot 89^{3} + \left(74 a + 83\right)\cdot 89^{4} + \left(77 a + 19\right)\cdot 89^{5} + \left(61 a + 80\right)\cdot 89^{6} + \left(77 a + 32\right)\cdot 89^{7} + \left(3 a + 85\right)\cdot 89^{8} +O\left(89^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,6,7,3)$ |
| $(4,5)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(2,7)$ |
| $(3,6)$ |
| $(1,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,8)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(4,5)$ |
$2$ |
| $4$ |
$2$ |
$(2,7)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,3)(4,5)(6,7)$ |
$-2$ |
| $4$ |
$2$ |
$(1,8)(2,7)(3,6)$ |
$-2$ |
| $4$ |
$2$ |
$(2,3)(6,7)$ |
$2$ |
| $8$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $8$ |
$2$ |
$(1,8)(2,3)(6,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,8)(2,6,7,3)(4,5)$ |
$-2$ |
| $4$ |
$4$ |
$(1,5,8,4)(2,3,7,6)$ |
$0$ |
| $4$ |
$4$ |
$(2,3,7,6)$ |
$2$ |
| $8$ |
$4$ |
$(1,4,8,5)(3,6)$ |
$0$ |
| $8$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
| $8$ |
$4$ |
$(1,5)(2,6,7,3)(4,8)$ |
$0$ |
| $16$ |
$4$ |
$(1,3)(2,4,7,5)(6,8)$ |
$0$ |
| $16$ |
$4$ |
$(1,2,4,6)(3,8,7,5)$ |
$0$ |
| $16$ |
$8$ |
$(1,6,5,2,8,3,4,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
