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:
| $r_{ 1 }$ |
$=$ |
$ 8 + 31\cdot 53 + 22\cdot 53^{2} + 9\cdot 53^{3} + 52\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 4 + \left(23 a + 8\right)\cdot 53 + \left(26 a + 27\right)\cdot 53^{2} + \left(18 a + 24\right)\cdot 53^{3} + 44 a\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 8\cdot 53 + 8\cdot 53^{2} + 2\cdot 53^{3} + 5\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 14 + \left(23 a + 40\right)\cdot 53 + \left(36 a + 40\right)\cdot 53^{2} + \left(30 a + 8\right)\cdot 53^{3} + \left(45 a + 1\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 42 + \left(29 a + 13\right)\cdot 53 + \left(26 a + 3\right)\cdot 53^{2} + \left(34 a + 19\right)\cdot 53^{3} + 8 a\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 4 + \left(29 a + 4\right)\cdot 53 + \left(16 a + 4\right)\cdot 53^{2} + \left(22 a + 42\right)\cdot 53^{3} + \left(7 a + 46\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,6)$ |
| $(1,3)(2,6)(4,5)$ |
| $(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,5,2)(3,4,6)$ | $-1$ |
| $2$ | $6$ | $(1,4,2,3,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
