Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 24 + \left(50 a + 10\right)\cdot 53 + \left(3 a + 4\right)\cdot 53^{2} + \left(2 a + 41\right)\cdot 53^{3} + \left(2 a + 10\right)\cdot 53^{4} + \left(6 a + 23\right)\cdot 53^{5} + \left(10 a + 50\right)\cdot 53^{6} + \left(45 a + 16\right)\cdot 53^{7} + \left(45 a + 40\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 4\cdot 53 + 52\cdot 53^{2} + 24\cdot 53^{3} + 7\cdot 53^{4} + 7\cdot 53^{5} + 39\cdot 53^{6} + 14\cdot 53^{7} + 26\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 a + 32 + \left(2 a + 49\right)\cdot 53 + \left(49 a + 22\right)\cdot 53^{2} + \left(50 a + 45\right)\cdot 53^{3} + \left(50 a + 16\right)\cdot 53^{4} + \left(46 a + 45\right)\cdot 53^{5} + \left(42 a + 31\right)\cdot 53^{6} + \left(7 a + 28\right)\cdot 53^{7} + \left(7 a + 19\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 + 42\cdot 53 + 25\cdot 53^{2} + 13\cdot 53^{3} + 42\cdot 53^{4} + 13\cdot 53^{5} + 22\cdot 53^{6} + 26\cdot 53^{7} + 16\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 34 + \left(48 a + 11\right)\cdot 53 + \left(30 a + 42\right)\cdot 53^{2} + \left(20 a + 17\right)\cdot 53^{3} + \left(19 a + 12\right)\cdot 53^{4} + \left(52 a + 19\right)\cdot 53^{5} + \left(22 a + 14\right)\cdot 53^{6} + \left(38 a + 50\right)\cdot 53^{7} + \left(41 a + 16\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 a + 50 + \left(4 a + 40\right)\cdot 53 + \left(22 a + 11\right)\cdot 53^{2} + \left(32 a + 16\right)\cdot 53^{3} + \left(33 a + 16\right)\cdot 53^{4} + 50\cdot 53^{5} + 30 a\cdot 53^{6} + \left(14 a + 22\right)\cdot 53^{7} + \left(11 a + 39\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(2,4)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(2,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(2,3,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,5)(2,3,4,6)$ | $1$ |
| $8$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
