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:
| $r_{ 1 }$ |
$=$ |
$ 8 + 50\cdot 53 + 7\cdot 53^{2} + 50\cdot 53^{3} + 33\cdot 53^{4} + 15\cdot 53^{5} + 7\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 a + 4 + \left(33 a + 17\right)\cdot 53 + 23\cdot 53^{2} + \left(32 a + 4\right)\cdot 53^{3} + \left(5 a + 36\right)\cdot 53^{4} + \left(12 a + 1\right)\cdot 53^{5} + \left(46 a + 37\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 6 + \left(44 a + 23\right)\cdot 53 + \left(7 a + 21\right)\cdot 53^{2} + \left(42 a + 38\right)\cdot 53^{3} + \left(13 a + 24\right)\cdot 53^{4} + \left(23 a + 35\right)\cdot 53^{5} + \left(19 a + 4\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 45 + \left(19 a + 49\right)\cdot 53 + \left(52 a + 44\right)\cdot 53^{2} + \left(20 a + 25\right)\cdot 53^{3} + \left(47 a + 26\right)\cdot 53^{4} + \left(40 a + 44\right)\cdot 53^{5} + \left(6 a + 50\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 45 + \left(8 a + 18\right)\cdot 53 + \left(45 a + 8\right)\cdot 53^{2} + \left(10 a + 40\right)\cdot 53^{3} + \left(39 a + 37\right)\cdot 53^{4} + \left(29 a + 8\right)\cdot 53^{5} + \left(33 a + 6\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,5,3,4)$ |
| $(1,4)(2,3)$ |
| $(1,2,4,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $5$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $5$ | $4$ | $(1,4,2,5)$ | $0$ |
| $5$ | $4$ | $(1,5,2,4)$ | $0$ |
| $4$ | $5$ | $(1,2,5,3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
