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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 10 + 10\cdot 53 + \left(16 a + 13\right)\cdot 53^{2} + \left(19 a + 27\right)\cdot 53^{3} + \left(34 a + 44\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 a + 24 + \left(18 a + 41\right)\cdot 53 + \left(5 a + 32\right)\cdot 53^{2} + \left(10 a + 31\right)\cdot 53^{3} + \left(20 a + 45\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 44\cdot 53 + 40\cdot 53^{2} + 33\cdot 53^{3} + 9\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 39\cdot 53 + 12\cdot 53^{2} + 17\cdot 53^{3} + 45\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 45 + \left(34 a + 18\right)\cdot 53 + \left(47 a + 35\right)\cdot 53^{2} + \left(42 a + 13\right)\cdot 53^{3} + \left(32 a + 10\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 a + 34 + \left(52 a + 4\right)\cdot 53 + \left(36 a + 24\right)\cdot 53^{2} + \left(33 a + 35\right)\cdot 53^{3} + \left(18 a + 3\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
| $(1,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(3,6)$ | $0$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,3,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $1$ |
| $12$ | $6$ | $(2,4,5)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
