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:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 45 + \left(36 a + 43\right)\cdot 53 + \left(14 a + 33\right)\cdot 53^{2} + \left(41 a + 26\right)\cdot 53^{3} + \left(15 a + 21\right)\cdot 53^{4} + \left(20 a + 6\right)\cdot 53^{5} + \left(13 a + 20\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 49\cdot 53 + 49\cdot 53^{2} + 4\cdot 53^{3} + 45\cdot 53^{4} + 50\cdot 53^{5} + 8\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 a + 28 + \left(15 a + 10\right)\cdot 53 + \left(3 a + 51\right)\cdot 53^{2} + \left(4 a + 36\right)\cdot 53^{3} + \left(21 a + 29\right)\cdot 53^{4} + \left(51 a + 30\right)\cdot 53^{5} + \left(41 a + 42\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 6\cdot 53 + 25\cdot 53^{2} + 22\cdot 53^{3} + 15\cdot 53^{4} + 49\cdot 53^{5} + 33\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 43 + \left(16 a + 4\right)\cdot 53 + \left(38 a + 3\right)\cdot 53^{2} + \left(11 a + 18\right)\cdot 53^{3} + \left(37 a + 43\right)\cdot 53^{4} + \left(32 a + 18\right)\cdot 53^{5} + 39 a\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 38 + \left(37 a + 43\right)\cdot 53 + \left(49 a + 48\right)\cdot 53^{2} + \left(48 a + 49\right)\cdot 53^{3} + \left(31 a + 3\right)\cdot 53^{4} + \left(a + 3\right)\cdot 53^{5} + 11 a\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(3,6)$ |
| $(5,6)$ |
| $(1,2,5)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)(5,6)$ | $-1$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $6$ | $2$ | $(1,5)(3,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,5)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,6,3,5)$ | $1$ |
| $6$ | $4$ | $(1,6,3,5)(2,4)$ | $-1$ |
| $8$ | $6$ | $(1,6,4,3,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
