Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{2} + 126 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 95 a + 57 + \left(10 a + 13\right)\cdot 127 + \left(10 a + 29\right)\cdot 127^{2} + \left(23 a + 48\right)\cdot 127^{3} + \left(17 a + 2\right)\cdot 127^{4} + \left(2 a + 31\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 55\cdot 127 + 61\cdot 127^{2} + 79\cdot 127^{3} + 35\cdot 127^{4} + 7\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 a + 25 + \left(116 a + 56\right)\cdot 127 + \left(116 a + 28\right)\cdot 127^{2} + \left(103 a + 61\right)\cdot 127^{3} + \left(109 a + 123\right)\cdot 127^{4} + \left(124 a + 15\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 72 + 79\cdot 127 + 121\cdot 127^{2} + 51\cdot 127^{3} + 67\cdot 127^{4} + 51\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 94 a + 59 + \left(40 a + 51\right)\cdot 127 + \left(71 a + 118\right)\cdot 127^{2} + \left(33 a + 88\right)\cdot 127^{3} + \left(63 a + 124\right)\cdot 127^{4} + \left(3 a + 103\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 26 + \left(86 a + 125\right)\cdot 127 + \left(55 a + 21\right)\cdot 127^{2} + \left(93 a + 51\right)\cdot 127^{3} + \left(63 a + 27\right)\cdot 127^{4} + \left(123 a + 44\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$8$ |
$8$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
$0$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
