Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a + 47 + \left(20 a + 26\right)\cdot 59 + \left(50 a + 48\right)\cdot 59^{2} + \left(4 a + 27\right)\cdot 59^{3} + \left(21 a + 55\right)\cdot 59^{4} + \left(29 a + 12\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 11\cdot 59 + 28\cdot 59^{2} + 8\cdot 59^{3} + 10\cdot 59^{4} + 36\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 7 + \left(52 a + 9\right)\cdot 59 + \left(22 a + 30\right)\cdot 59^{2} + \left(40 a + 16\right)\cdot 59^{3} + \left(26 a + 31\right)\cdot 59^{4} + \left(51 a + 28\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 45\cdot 59 + 49\cdot 59^{2} + 48\cdot 59^{3} + 49\cdot 59^{4} + 24\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 a + 48 + \left(38 a + 45\right)\cdot 59 + \left(8 a + 19\right)\cdot 59^{2} + \left(54 a + 41\right)\cdot 59^{3} + \left(37 a + 12\right)\cdot 59^{4} + \left(29 a + 21\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 30 + \left(6 a + 38\right)\cdot 59 + 36 a\cdot 59^{2} + \left(18 a + 34\right)\cdot 59^{3} + \left(32 a + 17\right)\cdot 59^{4} + \left(7 a + 53\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6)(4,5)$ |
| $(1,2)(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,5)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,4,5)(2,6,3)$ | $-1$ |
| $2$ | $6$ | $(1,6,4,3,5,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
