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:
| $r_{ 1 }$ |
$=$ |
$ 41 a + 27 + \left(44 a + 14\right)\cdot 59 + \left(32 a + 5\right)\cdot 59^{2} + \left(27 a + 9\right)\cdot 59^{3} + \left(34 a + 36\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 9 + \left(14 a + 18\right)\cdot 59 + \left(26 a + 52\right)\cdot 59^{2} + \left(31 a + 3\right)\cdot 59^{3} + \left(24 a + 43\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 20\cdot 59 + 27\cdot 59^{2} + 32\cdot 59^{3} + 16\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 57 + 2\cdot 59 + \left(56 a + 57\right)\cdot 59^{2} + \left(a + 32\right)\cdot 59^{3} + \left(13 a + 36\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 a + 6 + \left(58 a + 54\right)\cdot 59 + \left(2 a + 53\right)\cdot 59^{2} + \left(57 a + 37\right)\cdot 59^{3} + \left(45 a + 47\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 + 7\cdot 59 + 40\cdot 59^{2} + 59^{3} + 56\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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $4$ | $3$ | $(1,2,3)$ | $-2$ |
| $18$ | $4$ | $(1,4)(2,6,3,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $-1$ |
| $12$ | $6$ | $(2,3)(4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
