Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 36 + \left(5 a + 16\right)\cdot 41 + \left(11 a + 35\right)\cdot 41^{2} + \left(15 a + 28\right)\cdot 41^{3} + \left(40 a + 34\right)\cdot 41^{4} + \left(39 a + 17\right)\cdot 41^{5} + \left(18 a + 35\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 11\cdot 41 + 35\cdot 41^{2} + 4\cdot 41^{3} + 37\cdot 41^{4} + 24\cdot 41^{5} + 23\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 6 + \left(35 a + 3\right)\cdot 41 + \left(29 a + 22\right)\cdot 41^{2} + \left(25 a + 22\right)\cdot 41^{3} + 17\cdot 41^{4} + \left(a + 15\right)\cdot 41^{5} + \left(22 a + 11\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 + 32\cdot 41 + 15\cdot 41^{2} + 35\cdot 41^{3} + 10\cdot 41^{4} + 18\cdot 41^{5} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 22 + \left(35 a + 22\right)\cdot 41 + 29 a\cdot 41^{2} + \left(25 a + 33\right)\cdot 41^{3} + 2\cdot 41^{4} + \left(a + 22\right)\cdot 41^{5} + \left(22 a + 34\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 a + 11 + \left(5 a + 36\right)\cdot 41 + \left(11 a + 13\right)\cdot 41^{2} + \left(15 a + 39\right)\cdot 41^{3} + \left(40 a + 19\right)\cdot 41^{4} + \left(39 a + 24\right)\cdot 41^{5} + \left(18 a + 17\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,4)$ |
| $(1,2)(3,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,4,6,3,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
