Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 49\cdot 83 + 5\cdot 83^{2} + 61\cdot 83^{3} + 24\cdot 83^{4} + 75\cdot 83^{5} + 2\cdot 83^{6} + 10\cdot 83^{7} + 45\cdot 83^{8} + 82\cdot 83^{9} + 48\cdot 83^{10} + 48\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 74 + \left(63 a + 47\right)\cdot 83 + \left(22 a + 53\right)\cdot 83^{2} + \left(32 a + 40\right)\cdot 83^{3} + \left(68 a + 3\right)\cdot 83^{4} + \left(61 a + 65\right)\cdot 83^{5} + \left(30 a + 41\right)\cdot 83^{6} + \left(45 a + 54\right)\cdot 83^{7} + \left(79 a + 64\right)\cdot 83^{8} + \left(51 a + 81\right)\cdot 83^{9} + \left(53 a + 62\right)\cdot 83^{10} + \left(7 a + 81\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 5\cdot 83 + 54\cdot 83^{2} + 2\cdot 83^{3} + 6\cdot 83^{4} + 55\cdot 83^{5} + 30\cdot 83^{6} + 67\cdot 83^{7} + 23\cdot 83^{8} + 77\cdot 83^{9} + 27\cdot 83^{10} + 45\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 a + 42 + \left(58 a + 52\right)\cdot 83 + \left(76 a + 10\right)\cdot 83^{2} + \left(56 a + 57\right)\cdot 83^{3} + \left(9 a + 69\right)\cdot 83^{4} + \left(27 a + 71\right)\cdot 83^{5} + \left(33 a + 36\right)\cdot 83^{6} + \left(49 a + 57\right)\cdot 83^{7} + \left(27 a + 60\right)\cdot 83^{8} + \left(16 a + 23\right)\cdot 83^{9} + \left(4 a + 28\right)\cdot 83^{10} + \left(39 a + 1\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 46 a + 79 + \left(24 a + 73\right)\cdot 83 + \left(6 a + 28\right)\cdot 83^{2} + \left(26 a + 37\right)\cdot 83^{3} + \left(73 a + 22\right)\cdot 83^{4} + \left(55 a + 6\right)\cdot 83^{5} + \left(49 a + 43\right)\cdot 83^{6} + \left(33 a + 73\right)\cdot 83^{7} + \left(55 a + 38\right)\cdot 83^{8} + \left(66 a + 12\right)\cdot 83^{9} + \left(78 a + 16\right)\cdot 83^{10} + \left(43 a + 36\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 75 a + 82 + \left(19 a + 19\right)\cdot 83 + \left(60 a + 13\right)\cdot 83^{2} + \left(50 a + 50\right)\cdot 83^{3} + \left(14 a + 39\right)\cdot 83^{4} + \left(21 a + 58\right)\cdot 83^{5} + \left(52 a + 10\right)\cdot 83^{6} + \left(37 a + 69\right)\cdot 83^{7} + \left(3 a + 15\right)\cdot 83^{8} + \left(31 a + 54\right)\cdot 83^{9} + \left(29 a + 64\right)\cdot 83^{10} + \left(75 a + 35\right)\cdot 83^{11} +O\left(83^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(3,6)$ |
| $(1,3)$ |
| $(1,5,2)(3,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(5,6)$ | $-1$ |
| $6$ | $2$ | $(2,5)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,5,2)(3,6,4)$ | $0$ |
| $6$ | $4$ | $(1,6,3,5)$ | $-1$ |
| $6$ | $4$ | $(1,3)(2,5,4,6)$ | $1$ |
| $8$ | $6$ | $(1,6,4,3,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
