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:
| $r_{ 1 }$ |
$=$ |
$ 50 a + 24 + \left(33 a + 38\right)\cdot 53 + \left(51 a + 26\right)\cdot 53^{2} + \left(27 a + 12\right)\cdot 53^{3} + \left(38 a + 19\right)\cdot 53^{4} + \left(32 a + 38\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 26 + \left(42 a + 52\right)\cdot 53 + \left(23 a + 31\right)\cdot 53^{2} + \left(30 a + 6\right)\cdot 53^{3} + \left(41 a + 37\right)\cdot 53^{4} + \left(12 a + 3\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a + 16 + \left(29 a + 6\right)\cdot 53 + \left(6 a + 42\right)\cdot 53^{2} + \left(48 a + 20\right)\cdot 53^{3} + \left(44 a + 38\right)\cdot 53^{4} + \left(47 a + 45\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 33 + \left(10 a + 47\right)\cdot 53 + \left(29 a + 31\right)\cdot 53^{2} + \left(22 a + 51\right)\cdot 53^{3} + \left(11 a + 13\right)\cdot 53^{4} + \left(40 a + 13\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 51 + \left(23 a + 48\right)\cdot 53 + \left(46 a + 38\right)\cdot 53^{2} + \left(4 a + 47\right)\cdot 53^{3} + \left(8 a + 10\right)\cdot 53^{4} + \left(5 a + 33\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 12 + \left(19 a + 18\right)\cdot 53 + \left(a + 40\right)\cdot 53^{2} + \left(25 a + 19\right)\cdot 53^{3} + \left(14 a + 39\right)\cdot 53^{4} + \left(20 a + 24\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(2,5,6)$ |
| $(1,6,4,2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,4,3)(2,5,6)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,3,4)(2,6,5)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,4,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,3,4)(2,5,6)$ | $-1$ |
| $3$ | $6$ | $(1,6,4,2,3,5)$ | $0$ |
| $3$ | $6$ | $(1,5,3,2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
