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:
| $r_{ 1 }$ |
$=$ |
$ 41 a + 41 + \left(11 a + 35\right)\cdot 53 + \left(3 a + 35\right)\cdot 53^{2} + \left(24 a + 18\right)\cdot 53^{3} + \left(19 a + 52\right)\cdot 53^{4} + \left(3 a + 35\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a + 15 + \left(34 a + 16\right)\cdot 53 + \left(43 a + 16\right)\cdot 53^{2} + \left(25 a + 49\right)\cdot 53^{3} + \left(22 a + 48\right)\cdot 53^{4} + \left(9 a + 46\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 2\cdot 53 + 28\cdot 53^{2} + 12\cdot 53^{3} + 15\cdot 53^{4} + 33\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 31\cdot 53 + 44\cdot 53^{2} + 16\cdot 53^{3} + 35\cdot 53^{4} + 3\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 a + 26 + \left(18 a + 31\right)\cdot 53 + \left(9 a + 50\right)\cdot 53^{2} + \left(27 a + 2\right)\cdot 53^{3} + \left(30 a + 7\right)\cdot 53^{4} + \left(43 a + 9\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a + 46 + \left(41 a + 41\right)\cdot 53 + \left(49 a + 36\right)\cdot 53^{2} + \left(28 a + 5\right)\cdot 53^{3} + 33 a\cdot 53^{4} + \left(49 a + 30\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,5)$ |
| $(1,3,2)(4,5,6)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,4,6,3)$ | $-1$ |
| $6$ | $4$ | $(1,4,6,3)(2,5)$ | $1$ |
| $8$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
