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:
| $r_{ 1 }$ |
$=$ |
$ 52 a + 2 + 12 a\cdot 53 + \left(22 a + 15\right)\cdot 53^{2} + \left(46 a + 24\right)\cdot 53^{3} + \left(13 a + 48\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 15 + \left(15 a + 5\right)\cdot 53 + \left(21 a + 18\right)\cdot 53^{2} + \left(28 a + 33\right)\cdot 53^{3} + \left(5 a + 29\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 a + 14 + \left(28 a + 18\right)\cdot 53 + \left(30 a + 6\right)\cdot 53^{2} + \left(51 a + 18\right)\cdot 53^{3} + \left(30 a + 43\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 51 + \left(40 a + 52\right)\cdot 53 + \left(30 a + 37\right)\cdot 53^{2} + \left(6 a + 28\right)\cdot 53^{3} + \left(39 a + 4\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 38 + \left(37 a + 47\right)\cdot 53 + \left(31 a + 34\right)\cdot 53^{2} + \left(24 a + 19\right)\cdot 53^{3} + \left(47 a + 23\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 39 + \left(24 a + 34\right)\cdot 53 + \left(22 a + 46\right)\cdot 53^{2} + \left(a + 34\right)\cdot 53^{3} + \left(22 a + 9\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.
