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:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 25 + \left(30 a + 50\right)\cdot 53 + \left(31 a + 50\right)\cdot 53^{2} + \left(35 a + 50\right)\cdot 53^{3} + \left(29 a + 9\right)\cdot 53^{4} + \left(10 a + 45\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 24\cdot 53 + 20\cdot 53^{2} + 44\cdot 53^{3} + 26\cdot 53^{4} + 9\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 3 + \left(35 a + 27\right)\cdot 53 + \left(11 a + 37\right)\cdot 53^{2} + \left(50 a + 49\right)\cdot 53^{3} + \left(46 a + 11\right)\cdot 53^{4} + \left(10 a + 36\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 18 + \left(17 a + 45\right)\cdot 53 + \left(41 a + 48\right)\cdot 53^{2} + \left(2 a + 26\right)\cdot 53^{3} + \left(6 a + 43\right)\cdot 53^{4} + \left(42 a + 32\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 3\cdot 53 + 13\cdot 53^{2} + 37\cdot 53^{3} + 26\cdot 53^{4} + 30\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 a + 41 + \left(22 a + 7\right)\cdot 53 + \left(21 a + 41\right)\cdot 53^{2} + \left(17 a + 2\right)\cdot 53^{3} + \left(23 a + 40\right)\cdot 53^{4} + \left(42 a + 4\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,2)(5,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,4,5,3)$ | $-1$ |
| $8$ | $6$ | $(1,3,2,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
