Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 21\cdot 83 + 11\cdot 83^{2} + 41\cdot 83^{3} + 78\cdot 83^{4} + 65\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 62 a + 6 + \left(81 a + 1\right)\cdot 83 + \left(9 a + 58\right)\cdot 83^{2} + \left(76 a + 61\right)\cdot 83^{3} + \left(20 a + 76\right)\cdot 83^{4} + \left(75 a + 38\right)\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 73\cdot 83 + 64\cdot 83^{2} + 10\cdot 83^{3} + 5\cdot 83^{4} + 17\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 39 + 53 a\cdot 83 + \left(82 a + 8\right)\cdot 83^{2} + \left(70 a + 51\right)\cdot 83^{3} + \left(26 a + 55\right)\cdot 83^{4} + \left(12 a + 65\right)\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 79 a + 43 + \left(29 a + 49\right)\cdot 83 + 37\cdot 83^{2} + \left(12 a + 39\right)\cdot 83^{3} + \left(56 a + 11\right)\cdot 83^{4} + \left(70 a + 51\right)\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 68 + \left(a + 20\right)\cdot 83 + \left(73 a + 69\right)\cdot 83^{2} + \left(6 a + 44\right)\cdot 83^{3} + \left(62 a + 21\right)\cdot 83^{4} + \left(7 a + 10\right)\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
