Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 13\cdot 53^{2} + 25\cdot 53^{3} + 13\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a + 4 + \left(35 a + 44\right)\cdot 53 + \left(40 a + 40\right)\cdot 53^{2} + \left(44 a + 52\right)\cdot 53^{3} + \left(11 a + 16\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 4 + \left(33 a + 48\right)\cdot 53 + \left(21 a + 2\right)\cdot 53^{2} + \left(14 a + 18\right)\cdot 53^{3} + \left(37 a + 38\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 10 + \left(17 a + 52\right)\cdot 53 + \left(12 a + 8\right)\cdot 53^{2} + \left(8 a + 32\right)\cdot 53^{3} + \left(41 a + 19\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 52\cdot 53 + 36\cdot 53^{2} + 29\cdot 53^{3} + 3\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 a + 32 + \left(19 a + 14\right)\cdot 53 + \left(31 a + 3\right)\cdot 53^{2} + \left(38 a + 1\right)\cdot 53^{3} + \left(15 a + 14\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$2$ |
| $6$ |
$2$ |
$(2,4)$ |
$0$ |
| $9$ |
$2$ |
$(2,4)(5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,4)$ |
$-2$ |
| $4$ |
$3$ |
$(1,2,4)(3,5,6)$ |
$1$ |
| $18$ |
$4$ |
$(1,3)(2,6,4,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,2,6,4,3)$ |
$-1$ |
| $12$ |
$6$ |
$(2,4)(3,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
