Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 16 + \left(40 a + 27\right)\cdot 59 + \left(18 a + 33\right)\cdot 59^{2} + \left(7 a + 28\right)\cdot 59^{4} + \left(42 a + 41\right)\cdot 59^{5} + \left(20 a + 17\right)\cdot 59^{6} + \left(12 a + 30\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 44\cdot 59 + 10\cdot 59^{2} + 9\cdot 59^{3} + 57\cdot 59^{5} + 28\cdot 59^{6} + 23\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 50\cdot 59 + 19\cdot 59^{2} + 54\cdot 59^{3} + 35\cdot 59^{4} + 48\cdot 59^{5} + 43\cdot 59^{6} + 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 48 + \left(53 a + 58\right)\cdot 59 + \left(52 a + 50\right)\cdot 59^{2} + \left(20 a + 51\right)\cdot 59^{3} + \left(6 a + 16\right)\cdot 59^{4} + \left(26 a + 55\right)\cdot 59^{5} + \left(34 a + 40\right)\cdot 59^{6} + \left(6 a + 4\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 a + 24 + \left(5 a + 18\right)\cdot 59 + \left(6 a + 50\right)\cdot 59^{2} + \left(38 a + 19\right)\cdot 59^{3} + \left(52 a + 2\right)\cdot 59^{4} + \left(32 a + 16\right)\cdot 59^{5} + \left(24 a + 49\right)\cdot 59^{6} + \left(52 a + 35\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 47 + \left(18 a + 36\right)\cdot 59 + \left(40 a + 11\right)\cdot 59^{2} + \left(58 a + 41\right)\cdot 59^{3} + \left(51 a + 34\right)\cdot 59^{4} + \left(16 a + 17\right)\cdot 59^{5} + \left(38 a + 55\right)\cdot 59^{6} + \left(46 a + 21\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)$ |
| $(1,2)(3,6)$ |
| $(1,2,4)(3,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,6)(2,3)$ |
$-1$ |
| $3$ |
$2$ |
$(1,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,2)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,4)(3,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,2,6,3)$ |
$-1$ |
| $6$ |
$4$ |
$(1,5,6,4)(2,3)$ |
$1$ |
| $8$ |
$6$ |
$(1,3,5,6,2,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
