Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{3} + 3 x + 99 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 47 a^{2} + 71 a + 69 + \left(45 a^{2} + 67 a + 10\right)\cdot 101 + \left(37 a^{2} + 64 a + 2\right)\cdot 101^{2} + \left(56 a^{2} + 9 a + 41\right)\cdot 101^{3} + \left(84 a^{2} + 53 a + 84\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 94 a^{2} + 17 a + 2 + \left(73 a^{2} + 13 a + 84\right)\cdot 101 + \left(20 a^{2} + 49 a + 61\right)\cdot 101^{2} + \left(45 a^{2} + 100 a + 67\right)\cdot 101^{3} + \left(34 a^{2} + 19 a + 48\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 a^{2} + 77 a + 90 + \left(13 a + 37\right)\cdot 101 + \left(27 a^{2} + 85 a + 74\right)\cdot 101^{2} + \left(83 a^{2} + 45 a + 42\right)\cdot 101^{3} + \left(50 a^{2} + 46 a + 81\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 75 a^{2} + 82 a + 93 + \left(17 a^{2} + 88 a + 44\right)\cdot 101 + \left(86 a^{2} + 94 a + 56\right)\cdot 101^{2} + \left(9 a^{2} + 22 a + 80\right)\cdot 101^{3} + \left(49 a^{2} + 10 a + 34\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a^{2} + 24 a + 96 + \left(46 a^{2} + 90 a + 11\right)\cdot 101 + \left(91 a^{2} + 37 a + 9\right)\cdot 101^{2} + \left(81 a^{2} + 91 a + 92\right)\cdot 101^{3} + \left(46 a^{2} + 97 a + 8\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 a^{2} + 6 a + 63 + \left(9 a^{2} + 44 a + 39\right)\cdot 101 + \left(73 a^{2} + 99 a + 73\right)\cdot 101^{2} + \left(63 a^{2} + 100 a + 55\right)\cdot 101^{3} + \left(70 a^{2} + 50 a + 56\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 88 a^{2} + 54 a + 18 + \left(48 a^{2} + 81 a + 6\right)\cdot 101 + \left(26 a^{2} + a + 38\right)\cdot 101^{2} + \left(12 a^{2} + 61 a + 85\right)\cdot 101^{3} + \left(42 a^{2} + 94 a + 20\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 39 a^{2} + 66 a + 21 + \left(34 a^{2} + 31 a + 78\right)\cdot 101 + \left(89 a^{2} + 4 a + 62\right)\cdot 101^{2} + \left(78 a^{2} + 17 a + 16\right)\cdot 101^{3} + \left(9 a^{2} + 97 a + 57\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 71 a^{2} + 7 a + 57 + \left(26 a^{2} + 74 a + 90\right)\cdot 101 + \left(53 a^{2} + 67 a + 25\right)\cdot 101^{2} + \left(73 a^{2} + 55 a + 23\right)\cdot 101^{3} + \left(15 a^{2} + 34 a + 11\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,7,2,6,4,9,5,8,3)$ |
| $(1,9)(2,6)(3,5)(4,7)$ |
| $(1,6,5)(2,9,3)(4,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,9)(2,6)(3,5)(4,7)$ | $0$ |
| $2$ | $3$ | $(1,6,5)(2,9,3)(4,8,7)$ | $-1$ |
| $2$ | $9$ | $(1,7,2,6,4,9,5,8,3)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,2,4,5,3,7,6,9,8)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,4,3,6,8,2,5,7,9)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
