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:
| $r_{ 1 }$ |
$=$ |
$ 52 a + 2 + \left(11 a + 2\right)\cdot 53 + \left(5 a + 22\right)\cdot 53^{2} + \left(28 a + 52\right)\cdot 53^{3} + \left(41 a + 36\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 a + 12 + \left(33 a + 35\right)\cdot 53 + \left(22 a + 24\right)\cdot 53^{2} + \left(38 a + 40\right)\cdot 53^{3} + \left(48 a + 27\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 46\cdot 53 + 26\cdot 53^{2} + 44\cdot 53^{3} + 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 51 + \left(41 a + 50\right)\cdot 53 + \left(47 a + 30\right)\cdot 53^{2} + 24 a\cdot 53^{3} + \left(11 a + 16\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 41 + \left(19 a + 17\right)\cdot 53 + \left(30 a + 28\right)\cdot 53^{2} + \left(14 a + 12\right)\cdot 53^{3} + \left(4 a + 25\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 6\cdot 53 + 26\cdot 53^{2} + 8\cdot 53^{3} + 51\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,6)$ |
| $(2,5)(3,6)$ |
| $(1,6,5)(2,4,3)$ |
| $(1,2,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,6,5)(2,4,3)$ | $0$ |
| $6$ | $4$ | $(1,2,4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
