Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{2} + 103 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 52 a + 27 + \left(68 a + 75\right)\cdot 107 + \left(11 a + 52\right)\cdot 107^{2} + \left(3 a + 43\right)\cdot 107^{3} + \left(96 a + 53\right)\cdot 107^{4} + \left(33 a + 72\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 a + 105 + \left(25 a + 16\right)\cdot 107 + \left(84 a + 36\right)\cdot 107^{2} + \left(64 a + 31\right)\cdot 107^{3} + \left(96 a + 94\right)\cdot 107^{4} + \left(20 a + 40\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 2\cdot 107 + 22\cdot 107^{2} + 42\cdot 107^{3} + 36\cdot 107^{4} + 66\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 15 + \left(30 a + 56\right)\cdot 107 + \left(8 a + 102\right)\cdot 107^{2} + \left(41 a + 95\right)\cdot 107^{3} + \left(20 a + 84\right)\cdot 107^{4} + \left(75 a + 95\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 a + 21 + \left(38 a + 83\right)\cdot 107 + \left(95 a + 30\right)\cdot 107^{2} + \left(103 a + 44\right)\cdot 107^{3} + \left(10 a + 6\right)\cdot 107^{4} + \left(73 a + 5\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 a + 67 + \left(81 a + 74\right)\cdot 107 + \left(22 a + 26\right)\cdot 107^{2} + \left(42 a + 99\right)\cdot 107^{3} + \left(10 a + 94\right)\cdot 107^{4} + \left(86 a + 27\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 33 + 55\cdot 107 + 51\cdot 107^{2} + 33\cdot 107^{3} + 39\cdot 107^{4} + 64\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 102 a + 35 + \left(76 a + 64\right)\cdot 107 + \left(98 a + 105\right)\cdot 107^{2} + \left(65 a + 37\right)\cdot 107^{3} + \left(86 a + 18\right)\cdot 107^{4} + \left(31 a + 55\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,8)(3,7)(5,6)$ |
| $(2,3,5)(6,7,8)$ |
| $(1,3,4,7)(2,5,6,8)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
| $(1,5,4,8)(2,7,6,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
| $12$ | $2$ | $(2,8)(3,7)(5,6)$ | $0$ |
| $8$ | $3$ | $(2,3,5)(6,7,8)$ | $-1$ |
| $6$ | $4$ | $(1,3,4,7)(2,5,6,8)$ | $0$ |
| $8$ | $6$ | $(1,4)(2,7,5,6,3,8)$ | $1$ |
| $6$ | $8$ | $(1,5,3,6,4,8,7,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,8,3,2,4,5,7,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
