Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{3} + 3 x + 124 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 a^{2} + 59 a + 72 + \left(51 a^{2} + 14 a + 43\right)\cdot 127 + \left(33 a^{2} + 17 a + 125\right)\cdot 127^{2} + \left(104 a^{2} + 22 a + 85\right)\cdot 127^{3} + \left(55 a^{2} + 16 a + 58\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a^{2} + 50 a + 106 + \left(42 a^{2} + 13 a + 94\right)\cdot 127 + \left(57 a^{2} + 101 a + 19\right)\cdot 127^{2} + \left(39 a^{2} + 57 a\right)\cdot 127^{3} + \left(43 a^{2} + 105 a + 45\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 a^{2} + 124 + \left(117 a^{2} + 6 a + 48\right)\cdot 127 + \left(46 a^{2} + 112 a + 25\right)\cdot 127^{2} + \left(38 a^{2} + 122 a + 81\right)\cdot 127^{3} + \left(22 a^{2} + 3 a + 118\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 a^{2} + 63 a + 45 + \left(80 a^{2} + 34 a + 44\right)\cdot 127 + \left(30 a^{2} + 57 a + 93\right)\cdot 127^{2} + \left(85 a^{2} + 84 a + 91\right)\cdot 127^{3} + \left(63 a^{2} + 124 a + 85\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 a^{2} + 14 a + 59 + \left(4 a^{2} + 79 a + 19\right)\cdot 127 + \left(39 a^{2} + 95 a + 110\right)\cdot 127^{2} + \left(2 a^{2} + 111 a + 52\right)\cdot 127^{3} + \left(20 a^{2} + 23 a + 125\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 a^{2} + 2 a + 61 + \left(62 a^{2} + 35 a + 88\right)\cdot 127 + \left(6 a^{2} + 7 a + 91\right)\cdot 127^{2} + \left(115 a^{2} + 55 a + 92\right)\cdot 127^{3} + \left(30 a^{2} + 63 a + 71\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 67 a^{2} + 68 a + 45 + \left(85 a^{2} + 106 a + 112\right)\cdot 127 + \left(46 a^{2} + 124 a + 24\right)\cdot 127^{2} + \left(111 a^{2} + 108 a + 100\right)\cdot 127^{3} + \left(48 a^{2} + 106 a + 44\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 69 a^{2} + 125 a + 73 + \left(31 a^{2} + 102 a + 26\right)\cdot 127 + \left(52 a^{2} + 82 a + 56\right)\cdot 127^{2} + \left(18 a^{2} + 80 a + 26\right)\cdot 127^{3} + \left(51 a^{2} + 108 a + 112\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 122 a^{2} + 52 + \left(32 a^{2} + 116 a + 29\right)\cdot 127 + \left(68 a^{2} + 36 a + 88\right)\cdot 127^{2} + \left(120 a^{2} + 118 a + 103\right)\cdot 127^{3} + \left(44 a^{2} + 81 a + 99\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,6)(2,4)(3,8)(7,9)$ |
| $(1,7,3)(2,5,4)(6,8,9)$ |
| $(1,8,4,7,9,2,3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,6)(2,4)(3,8)(7,9)$ | $0$ |
| $2$ | $3$ | $(1,7,3)(2,5,4)(6,8,9)$ | $-1$ |
| $2$ | $9$ | $(1,8,4,7,9,2,3,6,5)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,4,9,3,5,8,7,2,6)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,9,5,7,6,4,3,8,2)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
