Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 + 16\cdot 53 + 17\cdot 53^{2} + 15\cdot 53^{3} + 33\cdot 53^{4} + 39\cdot 53^{5} + 39\cdot 53^{6} + 37\cdot 53^{7} + 36\cdot 53^{8} + 5\cdot 53^{9} + 17\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 a + 19 + \left(48 a + 2\right)\cdot 53 + \left(51 a + 2\right)\cdot 53^{2} + \left(12 a + 10\right)\cdot 53^{3} + \left(30 a + 45\right)\cdot 53^{4} + \left(51 a + 8\right)\cdot 53^{5} + \left(31 a + 35\right)\cdot 53^{6} + \left(26 a + 51\right)\cdot 53^{7} + \left(44 a + 52\right)\cdot 53^{8} + \left(48 a + 4\right)\cdot 53^{9} + \left(6 a + 17\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a + 37 + 48 a\cdot 53 + \left(51 a + 51\right)\cdot 53^{2} + \left(12 a + 42\right)\cdot 53^{3} + \left(30 a + 5\right)\cdot 53^{4} + \left(51 a + 27\right)\cdot 53^{5} + \left(31 a + 47\right)\cdot 53^{6} + \left(26 a + 32\right)\cdot 53^{7} + \left(44 a + 7\right)\cdot 53^{8} + \left(48 a + 3\right)\cdot 53^{9} + \left(6 a + 4\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 36\cdot 53 + 35\cdot 53^{2} + 37\cdot 53^{3} + 19\cdot 53^{4} + 13\cdot 53^{5} + 13\cdot 53^{6} + 15\cdot 53^{7} + 16\cdot 53^{8} + 47\cdot 53^{9} + 35\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 34 + \left(4 a + 50\right)\cdot 53 + \left(a + 50\right)\cdot 53^{2} + \left(40 a + 42\right)\cdot 53^{3} + \left(22 a + 7\right)\cdot 53^{4} + \left(a + 44\right)\cdot 53^{5} + \left(21 a + 17\right)\cdot 53^{6} + \left(26 a + 1\right)\cdot 53^{7} + 8 a\cdot 53^{8} + \left(4 a + 48\right)\cdot 53^{9} + \left(46 a + 35\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 16 + \left(4 a + 52\right)\cdot 53 + \left(a + 1\right)\cdot 53^{2} + \left(40 a + 10\right)\cdot 53^{3} + \left(22 a + 47\right)\cdot 53^{4} + \left(a + 25\right)\cdot 53^{5} + \left(21 a + 5\right)\cdot 53^{6} + \left(26 a + 20\right)\cdot 53^{7} + \left(8 a + 45\right)\cdot 53^{8} + \left(4 a + 49\right)\cdot 53^{9} + \left(46 a + 48\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(3,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,5,4,2)(3,6)$ | $1$ |
| $8$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
