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:
| $r_{ 1 }$ |
$=$ |
$ 35 a + 73 + \left(13 a + 3\right)\cdot 89 + \left(49 a + 16\right)\cdot 89^{2} + \left(2 a + 43\right)\cdot 89^{3} + \left(16 a + 58\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 + 28\cdot 89 + 19\cdot 89^{2} + 84\cdot 89^{3} + 19\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 a + 10 + \left(41 a + 26\right)\cdot 89 + \left(2 a + 81\right)\cdot 89^{2} + \left(51 a + 20\right)\cdot 89^{3} + \left(2 a + 47\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 25 + \left(47 a + 86\right)\cdot 89 + \left(86 a + 56\right)\cdot 89^{2} + \left(37 a + 19\right)\cdot 89^{3} + \left(86 a + 14\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 54 a + 51 + \left(75 a + 62\right)\cdot 89 + \left(39 a + 79\right)\cdot 89^{2} + \left(86 a + 11\right)\cdot 89^{3} + \left(72 a + 79\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 a + 84 + \left(71 a + 80\right)\cdot 89 + \left(31 a + 64\right)\cdot 89^{2} + \left(70 a + 35\right)\cdot 89^{3} + \left(31 a + 81\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 a + 47 + \left(17 a + 67\right)\cdot 89 + \left(57 a + 37\right)\cdot 89^{2} + \left(18 a + 51\right)\cdot 89^{3} + \left(57 a + 55\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
