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:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 27 + \left(4 a + 57\right)\cdot 59 + \left(29 a + 52\right)\cdot 59^{2} + \left(14 a + 2\right)\cdot 59^{3} + \left(2 a + 54\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 54 + 24\cdot 59 + 40\cdot 59^{2} + 46\cdot 59^{3} + 40\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 6 + \left(54 a + 24\right)\cdot 59 + \left(29 a + 18\right)\cdot 59^{2} + \left(44 a + 47\right)\cdot 59^{3} + \left(56 a + 41\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 51 + \left(40 a + 36\right)\cdot 59 + \left(17 a + 20\right)\cdot 59^{2} + \left(43 a + 52\right)\cdot 59^{3} + \left(29 a + 15\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 13 + \left(18 a + 56\right)\cdot 59 + \left(41 a + 56\right)\cdot 59^{2} + \left(15 a + 18\right)\cdot 59^{3} + \left(29 a + 2\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 + 36\cdot 59 + 46\cdot 59^{2} + 8\cdot 59^{3} + 22\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
| $(1,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(1,3)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $4$ | $3$ | $(2,4,5)$ | $-2$ |
| $18$ | $4$ | $(1,4,3,2)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,3,4,6,5)$ | $-1$ |
| $12$ | $6$ | $(1,3)(2,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
