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:
| $r_{ 1 }$ |
$=$ |
$ 43 a + 49 + \left(48 a + 39\right)\cdot 53 + \left(8 a + 52\right)\cdot 53^{2} + \left(11 a + 3\right)\cdot 53^{3} + \left(38 a + 49\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 47 + \left(49 a + 2\right)\cdot 53 + \left(46 a + 4\right)\cdot 53^{2} + \left(49 a + 3\right)\cdot 53^{3} + \left(2 a + 34\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 a + 42 + \left(3 a + 28\right)\cdot 53 + \left(6 a + 36\right)\cdot 53^{2} + \left(3 a + 49\right)\cdot 53^{3} + \left(50 a + 48\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 4\cdot 53 + 45\cdot 53^{2} + 10\cdot 53^{3} + 9\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 49\cdot 53 + 33\cdot 53^{2} + 51\cdot 53^{3} + 38\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 9 + \left(4 a + 33\right)\cdot 53 + \left(44 a + 39\right)\cdot 53^{2} + \left(41 a + 39\right)\cdot 53^{3} + \left(14 a + 31\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)$ |
| $(1,2,4)(3,5,6)$ |
| $(4,5)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,3)$ | $-1$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
| $4$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $4$ | $6$ | $(1,3,5,6,2,4)$ | $0$ |
| $4$ | $6$ | $(1,4,2,6,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
