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:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 33 + \left(10 a + 2\right)\cdot 59 + \left(2 a + 45\right)\cdot 59^{2} + \left(46 a + 43\right)\cdot 59^{3} + \left(8 a + 34\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 a + 35 + \left(38 a + 10\right)\cdot 59 + \left(3 a + 40\right)\cdot 59^{2} + \left(47 a + 20\right)\cdot 59^{3} + \left(9 a + 49\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 a + 57 + \left(48 a + 47\right)\cdot 59 + \left(56 a + 36\right)\cdot 59^{2} + \left(12 a + 28\right)\cdot 59^{3} + \left(50 a + 56\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 27 + \left(13 a + 9\right)\cdot 59 + \left(2 a + 1\right)\cdot 59^{2} + \left(45 a + 18\right)\cdot 59^{3} + \left(12 a + 28\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 6 + \left(45 a + 44\right)\cdot 59 + \left(56 a + 48\right)\cdot 59^{2} + \left(13 a + 1\right)\cdot 59^{3} + \left(46 a + 55\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 22 + \left(20 a + 3\right)\cdot 59 + \left(55 a + 5\right)\cdot 59^{2} + \left(11 a + 5\right)\cdot 59^{3} + \left(49 a + 12\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4,3)$ |
| $(1,2)(3,5)(4,6)$ |
| $(1,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(2,4,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,3,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,5)(2,3,4)$ | $-1$ |
| $3$ | $6$ | $(1,2,6,4,5,3)$ | $0$ |
| $3$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
