Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 69 a + 70 + \left(62 a + 81\right)\cdot 89 + \left(39 a + 25\right)\cdot 89^{2} + \left(59 a + 34\right)\cdot 89^{3} + \left(79 a + 62\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 5\cdot 89 + 11\cdot 89^{2} + 38\cdot 89^{3} + 66\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 81 + \left(9 a + 19\right)\cdot 89 + \left(20 a + 23\right)\cdot 89^{2} + \left(77 a + 51\right)\cdot 89^{3} + \left(20 a + 54\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 19 + \left(26 a + 7\right)\cdot 89 + \left(49 a + 63\right)\cdot 89^{2} + \left(29 a + 54\right)\cdot 89^{3} + \left(9 a + 26\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 72 + 83\cdot 89 + 77\cdot 89^{2} + 50\cdot 89^{3} + 22\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 74 a + 8 + \left(79 a + 69\right)\cdot 89 + \left(68 a + 65\right)\cdot 89^{2} + \left(11 a + 37\right)\cdot 89^{3} + \left(68 a + 34\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,2,4,5)$ |
| $(1,5,3)(2,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,5)(3,4)$ | $1$ |
| $8$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ |
| $6$ | $4$ | $(1,2,4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
