Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 15 + \left(27 a + 24\right)\cdot 53 + \left(28 a + 26\right)\cdot 53^{2} + \left(19 a + 42\right)\cdot 53^{3} + \left(2 a + 31\right)\cdot 53^{4} + \left(7 a + 33\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 + 40\cdot 53 + 53^{2} + 15\cdot 53^{3} + 13\cdot 53^{4} + 6\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 31\cdot 53 + 31\cdot 53^{2} + 8\cdot 53^{3} + 42\cdot 53^{4} + 9\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 a + 18 + \left(4 a + 10\right)\cdot 53 + \left(20 a + 8\right)\cdot 53^{2} + \left(37 a + 15\right)\cdot 53^{3} + \left(14 a + 14\right)\cdot 53^{4} + 29 a\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 48 + \left(48 a + 47\right)\cdot 53 + \left(32 a + 30\right)\cdot 53^{2} + \left(15 a + 38\right)\cdot 53^{3} + \left(38 a + 35\right)\cdot 53^{4} + \left(23 a + 49\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 5 + \left(25 a + 4\right)\cdot 53 + \left(24 a + 7\right)\cdot 53^{2} + \left(33 a + 39\right)\cdot 53^{3} + \left(50 a + 21\right)\cdot 53^{4} + \left(45 a + 6\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)(3,5,6)$ |
| $(2,3)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,6)$ |
$1$ |
| $3$ |
$2$ |
$(1,6)(2,3)$ |
$-1$ |
| $6$ |
$2$ |
$(2,4)(3,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,4)(3,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,3,6,2)$ |
$-1$ |
| $6$ |
$4$ |
$(1,5,6,4)(2,3)$ |
$1$ |
| $8$ |
$6$ |
$(1,3,5,6,2,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
