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 a + 35 + \left(25 a + 4\right)\cdot 53 + \left(49 a + 10\right)\cdot 53^{2} + \left(11 a + 34\right)\cdot 53^{3} + 35\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 21\cdot 53 + 16\cdot 53^{2} + 29\cdot 53^{3} + 35\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 52 + \left(27 a + 10\right)\cdot 53 + \left(3 a + 23\right)\cdot 53^{2} + \left(41 a + 32\right)\cdot 53^{3} + \left(52 a + 24\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 a + 51 + \left(41 a + 51\right)\cdot 53 + \left(20 a + 39\right)\cdot 53^{2} + \left(31 a + 47\right)\cdot 53^{3} + \left(25 a + 16\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 2\cdot 53 + 52\cdot 53^{2} + 10\cdot 53^{3} + 26\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 47 + \left(11 a + 8\right)\cdot 53 + \left(32 a + 28\right)\cdot 53^{2} + \left(21 a + 46\right)\cdot 53^{3} + \left(27 a + 34\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 36 + 5\cdot 53 + 42\cdot 53^{2} + 10\cdot 53^{3} + 38\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
