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:
| $r_{ 1 }$ |
$=$ |
$ 23 a + 44 + \left(76 a + 56\right)\cdot 83 + \left(70 a + 15\right)\cdot 83^{2} + \left(44 a + 66\right)\cdot 83^{3} + \left(6 a + 20\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 a + 81 + \left(82 a + 47\right)\cdot 83 + \left(45 a + 54\right)\cdot 83^{2} + \left(64 a + 67\right)\cdot 83^{3} + \left(42 a + 10\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 17\cdot 83 + 66\cdot 83^{2} + 79\cdot 83^{3} + 27\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 a + 36 + 9\cdot 83 + \left(37 a + 18\right)\cdot 83^{2} + \left(18 a + 3\right)\cdot 83^{3} + \left(40 a + 72\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 + 8\cdot 83 + 83^{2} + 75\cdot 83^{3} + 51\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 a + 67 + \left(6 a + 26\right)\cdot 83 + \left(12 a + 10\right)\cdot 83^{2} + \left(38 a + 40\right)\cdot 83^{3} + \left(76 a + 65\right)\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 |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
