Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 75 a + 87 + \left(5 a + 44\right)\cdot 89 + \left(56 a + 10\right)\cdot 89^{2} + \left(53 a + 31\right)\cdot 89^{3} + \left(75 a + 1\right)\cdot 89^{4} + \left(52 a + 13\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 45 + \left(22 a + 45\right)\cdot 89 + \left(20 a + 25\right)\cdot 89^{2} + \left(72 a + 17\right)\cdot 89^{3} + \left(69 a + 77\right)\cdot 89^{4} + \left(31 a + 83\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 86 a + 66 + \left(66 a + 18\right)\cdot 89 + \left(68 a + 56\right)\cdot 89^{2} + \left(16 a + 57\right)\cdot 89^{3} + \left(19 a + 48\right)\cdot 89^{4} + \left(57 a + 58\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 87\cdot 89 + 21\cdot 89^{2} + 12\cdot 89^{3} + 4\cdot 89^{4} + 38\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 78 + \left(83 a + 10\right)\cdot 89 + \left(32 a + 41\right)\cdot 89^{2} + \left(35 a + 83\right)\cdot 89^{3} + \left(13 a + 31\right)\cdot 89^{4} + \left(36 a + 40\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 + 59\cdot 89 + 22\cdot 89^{2} + 65\cdot 89^{3} + 14\cdot 89^{4} + 33\cdot 89^{5} +O\left(89^{ 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 values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$8$ |
$8$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
$0$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
