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 a + 34 + \left(30 a + 27\right)\cdot 53 + \left(13 a + 48\right)\cdot 53^{2} + \left(23 a + 30\right)\cdot 53^{3} + \left(47 a + 3\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 + 36\cdot 53 + 39\cdot 53^{2} + 11\cdot 53^{3} + 13\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 a + 45 + \left(22 a + 26\right)\cdot 53 + \left(39 a + 19\right)\cdot 53^{2} + \left(29 a + 4\right)\cdot 53^{3} + \left(5 a + 11\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 5\cdot 53 + 28\cdot 53^{2} + 26\cdot 53^{3} + 10\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 46 a + 45 + \left(44 a + 43\right)\cdot 53 + \left(31 a + 49\right)\cdot 53^{2} + \left(48 a + 40\right)\cdot 53^{3} + \left(40 a + 2\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 17 + \left(8 a + 18\right)\cdot 53 + \left(21 a + 26\right)\cdot 53^{2} + \left(4 a + 44\right)\cdot 53^{3} + \left(12 a + 11\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $15$ |
$2$ |
$(1,2)$ |
$-3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
