Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 65 + \left(15 a + 64\right)\cdot 83 + \left(58 a + 53\right)\cdot 83^{2} + \left(23 a + 19\right)\cdot 83^{3} + \left(18 a + 62\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 33 + \left(55 a + 25\right)\cdot 83 + \left(61 a + 17\right)\cdot 83^{2} + \left(66 a + 26\right)\cdot 83^{3} + \left(69 a + 56\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 79 a + 37 + \left(27 a + 76\right)\cdot 83 + \left(21 a + 23\right)\cdot 83^{2} + \left(16 a + 31\right)\cdot 83^{3} + \left(13 a + 59\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 a + 4 + \left(67 a + 58\right)\cdot 83 + \left(24 a + 13\right)\cdot 83^{2} + \left(59 a + 68\right)\cdot 83^{3} + \left(64 a + 56\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 21\cdot 83 + 3\cdot 83^{2} + 50\cdot 83^{3} + 79\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 + 3\cdot 83 + 54\cdot 83^{2} + 53\cdot 83^{3} + 17\cdot 83^{4} +O\left(83^{ 5 }\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 |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
