Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 44 + 26\cdot 53 + 7\cdot 53^{2} + 35\cdot 53^{3} + 12\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 52 + \left(21 a + 5\right)\cdot 53 + \left(20 a + 35\right)\cdot 53^{2} + \left(29 a + 36\right)\cdot 53^{3} + \left(4 a + 11\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 28 + \left(47 a + 5\right)\cdot 53 + 18\cdot 53^{2} + 9\cdot 53^{3} + \left(46 a + 34\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 8 + \left(31 a + 51\right)\cdot 53 + \left(32 a + 41\right)\cdot 53^{2} + \left(23 a + 27\right)\cdot 53^{3} + 48 a\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 + 48\cdot 53 + 28\cdot 53^{2} + 41\cdot 53^{3} + 40\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 a + 35 + \left(5 a + 20\right)\cdot 53 + \left(52 a + 27\right)\cdot 53^{2} + \left(52 a + 8\right)\cdot 53^{3} + \left(6 a + 6\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $4$ | $3$ | $(1,3,6)$ | $-2$ |
| $18$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $-1$ |
| $12$ | $6$ | $(1,3,6)(2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
