Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 36 a + 28 + \left(28 a + 15\right)\cdot 41 + \left(25 a + 37\right)\cdot 41^{2} + \left(9 a + 18\right)\cdot 41^{3} + \left(25 a + 28\right)\cdot 41^{4} + \left(20 a + 22\right)\cdot 41^{5} + \left(27 a + 30\right)\cdot 41^{6} + \left(17 a + 7\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 21 + \left(20 a + 23\right)\cdot 41 + \left(40 a + 31\right)\cdot 41^{2} + \left(9 a + 25\right)\cdot 41^{3} + \left(17 a + 40\right)\cdot 41^{4} + \left(13 a + 8\right)\cdot 41^{5} + \left(7 a + 16\right)\cdot 41^{6} + \left(28 a + 2\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 3\cdot 41 + 2\cdot 41^{2} + 20\cdot 41^{3} + 26\cdot 41^{4} + 22\cdot 41^{5} + 13\cdot 41^{6} + 17\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 13 + \left(12 a + 25\right)\cdot 41 + \left(15 a + 3\right)\cdot 41^{2} + \left(31 a + 22\right)\cdot 41^{3} + \left(15 a + 12\right)\cdot 41^{4} + \left(20 a + 18\right)\cdot 41^{5} + \left(13 a + 10\right)\cdot 41^{6} + \left(23 a + 33\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 20 + \left(20 a + 17\right)\cdot 41 + 9\cdot 41^{2} + \left(31 a + 15\right)\cdot 41^{3} + 23 a\cdot 41^{4} + \left(27 a + 32\right)\cdot 41^{5} + \left(33 a + 24\right)\cdot 41^{6} + \left(12 a + 38\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 37\cdot 41 + 38\cdot 41^{2} + 20\cdot 41^{3} + 14\cdot 41^{4} + 18\cdot 41^{5} + 27\cdot 41^{6} + 23\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6)(2,3,4)$ |
| $(1,2)(4,5)$ |
| $(1,5)(2,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,5,6)(2,3,4)$ | $0$ |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
