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:
| $r_{ 1 }$ |
$=$ |
$ 27 a + 23 + \left(34 a + 35\right)\cdot 53 + \left(48 a + 34\right)\cdot 53^{2} + \left(40 a + 27\right)\cdot 53^{3} + \left(13 a + 26\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 30\cdot 53 + 35\cdot 53^{2} + 41\cdot 53^{3} + 38\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 a + 9 + \left(32 a + 6\right)\cdot 53 + \left(35 a + 47\right)\cdot 53^{2} + \left(34 a + 42\right)\cdot 53^{3} + \left(25 a + 51\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 3\cdot 53 + 8\cdot 53^{2} + 23\cdot 53^{3} + 40\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 25 + \left(18 a + 40\right)\cdot 53 + \left(4 a + 35\right)\cdot 53^{2} + \left(12 a + 36\right)\cdot 53^{3} + \left(39 a + 40\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a + 14 + \left(20 a + 43\right)\cdot 53 + \left(17 a + 50\right)\cdot 53^{2} + \left(18 a + 39\right)\cdot 53^{3} + \left(27 a + 13\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
| $(1,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(1,2)$ |
$-2$ |
| $9$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$-2$ |
| $4$ |
$3$ |
$(3,4,6)$ |
$1$ |
| $18$ |
$4$ |
$(1,4,2,3)(5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,2,4,5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,2)(3,4,6)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
