Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 a + 7 + \left(25 a + 27\right)\cdot 59 + \left(55 a + 8\right)\cdot 59^{2} + \left(48 a + 12\right)\cdot 59^{3} + \left(22 a + 45\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 55\cdot 59 + 11\cdot 59^{2} + 41\cdot 59^{3} + 53\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 a + 24 + \left(33 a + 35\right)\cdot 59 + \left(3 a + 38\right)\cdot 59^{2} + \left(10 a + 5\right)\cdot 59^{3} + \left(36 a + 19\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 + 43\cdot 59 + 19\cdot 59^{2} + 52\cdot 59^{3} + 44\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 a + 6 + \left(53 a + 8\right)\cdot 59 + \left(23 a + 5\right)\cdot 59^{2} + \left(23 a + 33\right)\cdot 59^{3} + \left(23 a + 36\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 2 + \left(5 a + 7\right)\cdot 59 + \left(35 a + 34\right)\cdot 59^{2} + \left(35 a + 32\right)\cdot 59^{3} + \left(35 a + 36\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(4,5,6)$ |
| $(1,4)(2,5)(3,6)$ |
| $(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $2$ |
| $6$ | $2$ | $(1,2)$ | $0$ |
| $9$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $4$ | $3$ | $(1,2,3)$ | $-2$ |
| $18$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $-1$ |
| $12$ | $6$ | $(1,2)(4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
