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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 44 + \left(36 a + 21\right)\cdot 53 + \left(51 a + 47\right)\cdot 53^{2} + \left(22 a + 32\right)\cdot 53^{3} + \left(32 a + 52\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 3 + \left(15 a + 8\right)\cdot 53 + \left(49 a + 15\right)\cdot 53^{2} + \left(22 a + 5\right)\cdot 53^{3} + \left(38 a + 14\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 a + 12 + \left(31 a + 39\right)\cdot 53 + \left(50 a + 20\right)\cdot 53^{2} + \left(52 a + 25\right)\cdot 53^{3} + \left(5 a + 14\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 9 + \left(16 a + 31\right)\cdot 53 + \left(a + 5\right)\cdot 53^{2} + \left(30 a + 20\right)\cdot 53^{3} + 20 a\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 50 + \left(37 a + 44\right)\cdot 53 + \left(3 a + 37\right)\cdot 53^{2} + \left(30 a + 47\right)\cdot 53^{3} + \left(14 a + 38\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 41 + \left(21 a + 13\right)\cdot 53 + \left(2 a + 32\right)\cdot 53^{2} + 27\cdot 53^{3} + \left(47 a + 38\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,3)(2,6,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,3,4,5,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,5,4,3,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
