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:
| $r_{ 1 }$ |
$=$ |
$ 14 + 59\cdot 89 + 59\cdot 89^{2} + 81\cdot 89^{3} + 34\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 a + 17 + \left(83 a + 42\right)\cdot 89 + \left(48 a + 48\right)\cdot 89^{2} + \left(13 a + 21\right)\cdot 89^{3} + \left(64 a + 49\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 65 a + 84 + \left(44 a + 53\right)\cdot 89 + \left(66 a + 56\right)\cdot 89^{2} + 30\cdot 89^{3} + \left(62 a + 50\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 75 + 29\cdot 89 + 29\cdot 89^{2} + 7\cdot 89^{3} + 54\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 a + 72 + \left(5 a + 46\right)\cdot 89 + \left(40 a + 40\right)\cdot 89^{2} + \left(75 a + 67\right)\cdot 89^{3} + \left(24 a + 39\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 5 + \left(44 a + 35\right)\cdot 89 + \left(22 a + 32\right)\cdot 89^{2} + \left(88 a + 58\right)\cdot 89^{3} + \left(26 a + 38\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,6)$ |
| $(2,5)(3,6)$ |
| $(1,6,4,3)$ |
| $(1,3,5)(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,5)(3,4)$ | $1$ |
| $8$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
| $6$ | $4$ | $(1,6,4,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
