Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 a + 39 + \left(81 a + 17\right)\cdot 89 + \left(34 a + 9\right)\cdot 89^{2} + \left(66 a + 73\right)\cdot 89^{3} + \left(7 a + 10\right)\cdot 89^{4} + \left(74 a + 7\right)\cdot 89^{5} + \left(79 a + 35\right)\cdot 89^{6} +O\left(89^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 51 + \left(83 a + 63\right)\cdot 89 + \left(35 a + 75\right)\cdot 89^{2} + \left(10 a + 85\right)\cdot 89^{3} + \left(a + 46\right)\cdot 89^{4} + \left(55 a + 1\right)\cdot 89^{5} + \left(5 a + 44\right)\cdot 89^{6} +O\left(89^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 a + 54 + \left(7 a + 1\right)\cdot 89 + \left(54 a + 83\right)\cdot 89^{2} + \left(22 a + 57\right)\cdot 89^{3} + \left(81 a + 87\right)\cdot 89^{4} + \left(14 a + 72\right)\cdot 89^{5} + \left(9 a + 74\right)\cdot 89^{6} +O\left(89^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 + 35\cdot 89 + 67\cdot 89^{2} + 13\cdot 89^{3} + 37\cdot 89^{4} + 53\cdot 89^{5} + 51\cdot 89^{6} +O\left(89^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 45\cdot 89 + 54\cdot 89^{2} + 2\cdot 89^{3} + 40\cdot 89^{4} + 13\cdot 89^{5} + 33\cdot 89^{6} +O\left(89^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 82 a + 11 + \left(5 a + 15\right)\cdot 89 + \left(53 a + 66\right)\cdot 89^{2} + \left(78 a + 33\right)\cdot 89^{3} + \left(87 a + 44\right)\cdot 89^{4} + \left(33 a + 29\right)\cdot 89^{5} + \left(83 a + 28\right)\cdot 89^{6} +O\left(89^{ 7 }\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.
