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:
| $r_{ 1 }$ |
$=$ |
$ 28 a + 2 + \left(13 a + 48\right)\cdot 53 + \left(13 a + 40\right)\cdot 53^{2} + \left(41 a + 24\right)\cdot 53^{3} + \left(20 a + 31\right)\cdot 53^{4} + \left(52 a + 18\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 16 + \left(23 a + 41\right)\cdot 53 + \left(15 a + 25\right)\cdot 53^{2} + 30 a\cdot 53^{3} + \left(32 a + 50\right)\cdot 53^{4} + \left(30 a + 2\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 36\cdot 53 + 37\cdot 53^{2} + 10\cdot 53^{3} + 53^{4} + 39\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 48\cdot 53 + 15\cdot 53^{2} + 52\cdot 53^{3} + 11\cdot 53^{4} + 10\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a + 8 + \left(39 a + 21\right)\cdot 53 + \left(39 a + 27\right)\cdot 53^{2} + \left(11 a + 17\right)\cdot 53^{3} + \left(32 a + 20\right)\cdot 53^{4} + 48\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 a + 11 + \left(29 a + 16\right)\cdot 53 + \left(37 a + 11\right)\cdot 53^{2} + 22 a\cdot 53^{3} + \left(20 a + 44\right)\cdot 53^{4} + \left(22 a + 39\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,4,6)$ | $-1$ |
| $2$ | $6$ | $(1,4,5,6,3,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
