Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 50 + \left(50 a + 29\right)\cdot 53 + \left(32 a + 25\right)\cdot 53^{2} + \left(3 a + 15\right)\cdot 53^{3} + \left(43 a + 43\right)\cdot 53^{4} + \left(8 a + 32\right)\cdot 53^{5} + \left(21 a + 1\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 40\cdot 53 + 17\cdot 53^{2} + 52\cdot 53^{3} + 40\cdot 53^{4} + 10\cdot 53^{5} + 25\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 53 + 16\cdot 53^{2} + 20\cdot 53^{3} + 47\cdot 53^{4} + 24\cdot 53^{5} + 14\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 a + 32 + \left(13 a + 11\right)\cdot 53 + \left(43 a + 49\right)\cdot 53^{2} + \left(30 a + 49\right)\cdot 53^{3} + \left(52 a + 29\right)\cdot 53^{4} + \left(a + 28\right)\cdot 53^{5} + \left(16 a + 15\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 49 + \left(39 a + 22\right)\cdot 53 + \left(9 a + 49\right)\cdot 53^{2} + \left(22 a + 23\right)\cdot 53^{3} + 50\cdot 53^{4} + \left(51 a + 36\right)\cdot 53^{5} + \left(36 a + 24\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 24 + \left(2 a + 52\right)\cdot 53 + 20 a\cdot 53^{2} + \left(49 a + 50\right)\cdot 53^{3} + \left(9 a + 52\right)\cdot 53^{4} + \left(44 a + 24\right)\cdot 53^{5} + \left(31 a + 24\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2,5)(3,6,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $6$ | $2$ | $(2,5)(3,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,5)(3,6,4)$ | $0$ |
| $6$ | $4$ | $(1,3,4,2)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,6,3,5)$ | $-1$ |
| $8$ | $6$ | $(1,3,6,4,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
