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:
| $r_{ 1 }$ |
$=$ |
$ 37 + 60\cdot 89 + 86\cdot 89^{2} + 25\cdot 89^{3} + 16\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 84 + 17\cdot 89 + 25\cdot 89^{2} + 15\cdot 89^{3} + 30\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 a + 81 + \left(46 a + 22\right)\cdot 89 + \left(35 a + 39\right)\cdot 89^{2} + \left(27 a + 28\right)\cdot 89^{3} + \left(a + 10\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 63 + \left(42 a + 21\right)\cdot 89 + \left(53 a + 63\right)\cdot 89^{2} + \left(61 a + 6\right)\cdot 89^{3} + \left(87 a + 81\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 70 + 48\cdot 89 + 73\cdot 89^{2} + 31\cdot 89^{3} + 4\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 + 6\cdot 89 + 68\cdot 89^{2} + 69\cdot 89^{3} + 35\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $15$ | $2$ | $(1,2)$ | $-2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
