Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{3} + 3 x + 99 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 a^{2} + 32 a + 66 + \left(64 a^{2} + 35 a + 27\right)\cdot 101 + \left(69 a^{2} + 42 a + 38\right)\cdot 101^{2} + \left(69 a^{2} + 39 a + 38\right)\cdot 101^{3} + \left(49 a^{2} + 73 a + 99\right)\cdot 101^{4} + \left(66 a^{2} + 51 a + 31\right)\cdot 101^{5} + \left(99 a^{2} + 4 a + 98\right)\cdot 101^{6} + \left(30 a^{2} + 92 a + 61\right)\cdot 101^{7} + \left(2 a^{2} + a + 4\right)\cdot 101^{8} +O\left(101^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 a^{2} + 88 a + 33 + \left(21 a^{2} + 82 a + 43\right)\cdot 101 + \left(99 a^{2} + 68 a + 97\right)\cdot 101^{2} + \left(80 a^{2} + 98 a + 60\right)\cdot 101^{3} + \left(20 a^{2} + 90 a + 41\right)\cdot 101^{4} + \left(5 a^{2} + 100 a + 10\right)\cdot 101^{5} + \left(67 a^{2} + 86 a + 33\right)\cdot 101^{6} + \left(12 a^{2} + 30 a + 25\right)\cdot 101^{7} + \left(66 a^{2} + 30 a + 31\right)\cdot 101^{8} +O\left(101^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 81 a^{2} + 60 a + 61 + \left(27 a^{2} + 20 a + 55\right)\cdot 101 + \left(88 a^{2} + 6 a + 75\right)\cdot 101^{2} + \left(52 a^{2} + 9 a + 4\right)\cdot 101^{3} + \left(92 a^{2} + 99 a + 84\right)\cdot 101^{4} + \left(73 a^{2} + 16 a + 46\right)\cdot 101^{5} + \left(42 a^{2} + 61 a + 85\right)\cdot 101^{6} + \left(17 a^{2} + 99 a + 34\right)\cdot 101^{7} + \left(64 a^{2} + 52 a + 27\right)\cdot 101^{8} +O\left(101^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 88 a^{2} + 9 a + 75 + \left(8 a^{2} + 45 a + 17\right)\cdot 101 + \left(44 a^{2} + 52 a + 88\right)\cdot 101^{2} + \left(79 a^{2} + 52 a + 57\right)\cdot 101^{3} + \left(59 a^{2} + 29 a + 18\right)\cdot 101^{4} + \left(61 a^{2} + 32 a + 22\right)\cdot 101^{5} + \left(59 a^{2} + 35 a + 18\right)\cdot 101^{6} + \left(52 a^{2} + 10 a + 4\right)\cdot 101^{7} + \left(34 a^{2} + 46 a + 69\right)\cdot 101^{8} +O\left(101^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 + 27\cdot 101 + 93\cdot 101^{2} + 26\cdot 101^{3} + 97\cdot 101^{4} + 52\cdot 101^{5} + 25\cdot 101^{6} + 4\cdot 101^{7} + 45\cdot 101^{8} +O\left(101^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 94 + 13\cdot 101 + 43\cdot 101^{2} + 2\cdot 101^{3} + 8\cdot 101^{4} + 43\cdot 101^{5} + 66\cdot 101^{6} + 79\cdot 101^{7} + 66\cdot 101^{8} +O\left(101^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 95 a^{2} + 70 a + 89 + \left(68 a^{2} + 46 a + 36\right)\cdot 101 + \left(91 a^{2} + 97 a + 82\right)\cdot 101^{2} + \left(18 a^{2} + 47 a + 37\right)\cdot 101^{3} + \left(28 a^{2} + 80 a + 56\right)\cdot 101^{4} + \left(96 a^{2} + 96 a + 91\right)\cdot 101^{5} + \left(41 a^{2} + 13 a + 83\right)\cdot 101^{6} + \left(92 a^{2} + 56 a + 83\right)\cdot 101^{7} + \left(22 a^{2} + a + 45\right)\cdot 101^{8} +O\left(101^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 59\cdot 101 + 65\cdot 101^{2} + 71\cdot 101^{3} + 96\cdot 101^{4} + 4\cdot 101^{5} + 9\cdot 101^{6} + 17\cdot 101^{7} + 90\cdot 101^{8} +O\left(101^{ 9 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 40 a^{2} + 44 a + 80 + \left(10 a^{2} + 72 a + 20\right)\cdot 101 + \left(11 a^{2} + 35 a + 22\right)\cdot 101^{2} + \left(a^{2} + 55 a + 2\right)\cdot 101^{3} + \left(52 a^{2} + 30 a + 3\right)\cdot 101^{4} + \left(100 a^{2} + 4 a + 100\right)\cdot 101^{5} + \left(92 a^{2} + 84\right)\cdot 101^{6} + \left(96 a^{2} + 14 a + 92\right)\cdot 101^{7} + \left(11 a^{2} + 69 a + 23\right)\cdot 101^{8} +O\left(101^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,6)(3,7,8)(4,9,5)$ |
| $(2,7,9)(5,8,6)$ |
| $(2,6)(5,9)(7,8)$ |
| $(1,4,3)(2,9,7)(5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $9$ |
$2$ |
$(2,6)(5,9)(7,8)$ |
$1$ |
$1$ |
| $1$ |
$3$ |
$(1,4,3)(2,9,7)(5,8,6)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $1$ |
$3$ |
$(1,3,4)(2,7,9)(5,6,8)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $6$ |
$3$ |
$(1,2,6)(3,7,8)(4,9,5)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,7,6)(2,5,4)(3,9,8)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(2,7,9)(5,8,6)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,6,9)(2,3,8)(4,5,7)$ |
$0$ |
$0$ |
| $9$ |
$6$ |
$(1,4,3)(2,5,7,6,9,8)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $9$ |
$6$ |
$(1,3,4)(2,8,9,6,7,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
