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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 37\cdot 89 + 40\cdot 89^{2} + 85\cdot 89^{3} + 17\cdot 89^{4} + 86\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 + 41\cdot 89 + 21\cdot 89^{3} + 80\cdot 89^{4} + 17\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 83 a + 15 + \left(32 a + 75\right)\cdot 89 + \left(61 a + 54\right)\cdot 89^{2} + \left(27 a + 81\right)\cdot 89^{3} + \left(64 a + 77\right)\cdot 89^{4} + \left(40 a + 31\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 62 + \left(56 a + 44\right)\cdot 89 + \left(27 a + 6\right)\cdot 89^{2} + \left(61 a + 36\right)\cdot 89^{3} + \left(24 a + 55\right)\cdot 89^{4} + \left(48 a + 74\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a + 44 + \left(87 a + 54\right)\cdot 89 + \left(49 a + 26\right)\cdot 89^{2} + \left(30 a + 49\right)\cdot 89^{3} + \left(57 a + 18\right)\cdot 89^{4} + \left(32 a + 45\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 23 + \left(83 a + 63\right)\cdot 89 + \left(33 a + 69\right)\cdot 89^{2} + \left(50 a + 86\right)\cdot 89^{3} + \left(59 a + 30\right)\cdot 89^{4} + \left(77 a + 11\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 88 a + 51 + \left(a + 39\right)\cdot 89 + \left(39 a + 22\right)\cdot 89^{2} + \left(58 a + 35\right)\cdot 89^{3} + \left(31 a + 33\right)\cdot 89^{4} + \left(56 a + 38\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 a + 6 + 5 a\cdot 89 + \left(55 a + 46\right)\cdot 89^{2} + \left(38 a + 49\right)\cdot 89^{3} + \left(29 a + 41\right)\cdot 89^{4} + \left(11 a + 50\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,7,8,6)$ |
| $(1,5,7)(2,8,6)$ |
| $(1,8)(2,5)(6,7)$ |
| $(1,2)(3,4)(5,8)(6,7)$ |
| $(1,7,2,6)(3,5,4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,8)(6,7)$ | $-2$ |
| $12$ | $2$ | $(1,8)(2,5)(6,7)$ | $0$ |
| $8$ | $3$ | $(3,7,8)(4,6,5)$ | $-1$ |
| $6$ | $4$ | $(1,7,2,6)(3,5,4,8)$ | $0$ |
| $8$ | $6$ | $(1,7,3,2,6,4)(5,8)$ | $1$ |
| $6$ | $8$ | $(1,3,5,6,2,4,8,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,4,5,7,2,3,8,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
