Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 a + 25 + \left(26 a + 38\right)\cdot 53 + \left(47 a + 25\right)\cdot 53^{2} + \left(7 a + 47\right)\cdot 53^{3} + \left(46 a + 11\right)\cdot 53^{4} + \left(22 a + 43\right)\cdot 53^{5} + \left(49 a + 44\right)\cdot 53^{6} + \left(15 a + 48\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 15 + \left(16 a + 7\right)\cdot 53 + \left(31 a + 49\right)\cdot 53^{2} + \left(43 a + 41\right)\cdot 53^{3} + \left(10 a + 40\right)\cdot 53^{4} + \left(41 a + 26\right)\cdot 53^{5} + \left(20 a + 48\right)\cdot 53^{6} + \left(46 a + 38\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 38 + \left(26 a + 49\right)\cdot 53 + \left(5 a + 29\right)\cdot 53^{2} + \left(45 a + 31\right)\cdot 53^{3} + \left(6 a + 29\right)\cdot 53^{4} + \left(30 a + 35\right)\cdot 53^{5} + \left(3 a + 7\right)\cdot 53^{6} + \left(37 a + 10\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 33 + \left(36 a + 42\right)\cdot 53 + \left(21 a + 51\right)\cdot 53^{2} + \left(9 a + 25\right)\cdot 53^{3} + \left(42 a + 40\right)\cdot 53^{4} + \left(11 a + 21\right)\cdot 53^{5} + \left(32 a + 37\right)\cdot 53^{6} + \left(6 a + 44\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 28\cdot 53 + 35\cdot 53^{2} + 38\cdot 53^{3} + 49\cdot 53^{4} + 10\cdot 53^{5} + 45\cdot 53^{6} + 11\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 45\cdot 53 + 19\cdot 53^{2} + 26\cdot 53^{3} + 39\cdot 53^{4} + 20\cdot 53^{5} + 28\cdot 53^{6} + 4\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)$ |
| $(1,2)$ |
| $(1,3,5)(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(3,5)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
| $6$ | $4$ | $(1,4,2,3)$ | $-1$ |
| $6$ | $4$ | $(1,2)(3,6,4,5)$ | $1$ |
| $8$ | $6$ | $(1,4,6,2,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
