Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 16 + \left(7 a + 10\right)\cdot 41 + \left(8 a + 5\right)\cdot 41^{2} + \left(36 a + 27\right)\cdot 41^{3} + \left(36 a + 17\right)\cdot 41^{4} + \left(27 a + 15\right)\cdot 41^{5} + \left(31 a + 17\right)\cdot 41^{6} + \left(12 a + 2\right)\cdot 41^{7} + \left(35 a + 40\right)\cdot 41^{8} + \left(a + 19\right)\cdot 41^{9} + \left(36 a + 40\right)\cdot 41^{10} + \left(14 a + 36\right)\cdot 41^{11} + \left(39 a + 30\right)\cdot 41^{12} +O\left(41^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 + 31\cdot 41 + 2\cdot 41^{2} + 4\cdot 41^{3} + 14\cdot 41^{4} + 41^{5} + 22\cdot 41^{6} + 23\cdot 41^{7} + 32\cdot 41^{8} + 22\cdot 41^{9} + 23\cdot 41^{10} + 5\cdot 41^{11} +O\left(41^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 5 + \left(33 a + 22\right)\cdot 41 + \left(32 a + 22\right)\cdot 41^{2} + \left(4 a + 4\right)\cdot 41^{3} + \left(4 a + 10\right)\cdot 41^{4} + \left(13 a + 21\right)\cdot 41^{5} + \left(9 a + 2\right)\cdot 41^{6} + \left(28 a + 9\right)\cdot 41^{7} + \left(5 a + 10\right)\cdot 41^{8} + \left(39 a + 31\right)\cdot 41^{9} + \left(4 a + 23\right)\cdot 41^{10} + \left(26 a + 4\right)\cdot 41^{11} + \left(a + 11\right)\cdot 41^{12} +O\left(41^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a + 25 + \left(33 a + 30\right)\cdot 41 + \left(32 a + 35\right)\cdot 41^{2} + \left(4 a + 13\right)\cdot 41^{3} + \left(4 a + 23\right)\cdot 41^{4} + \left(13 a + 25\right)\cdot 41^{5} + \left(9 a + 23\right)\cdot 41^{6} + \left(28 a + 38\right)\cdot 41^{7} + 5 a\cdot 41^{8} + \left(39 a + 21\right)\cdot 41^{9} + 4 a\cdot 41^{10} + \left(26 a + 4\right)\cdot 41^{11} + \left(a + 10\right)\cdot 41^{12} +O\left(41^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 9\cdot 41 + 38\cdot 41^{2} + 36\cdot 41^{3} + 26\cdot 41^{4} + 39\cdot 41^{5} + 18\cdot 41^{6} + 17\cdot 41^{7} + 8\cdot 41^{8} + 18\cdot 41^{9} + 17\cdot 41^{10} + 35\cdot 41^{11} + 40\cdot 41^{12} +O\left(41^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 36 + \left(7 a + 18\right)\cdot 41 + \left(8 a + 18\right)\cdot 41^{2} + \left(36 a + 36\right)\cdot 41^{3} + \left(36 a + 30\right)\cdot 41^{4} + \left(27 a + 19\right)\cdot 41^{5} + \left(31 a + 38\right)\cdot 41^{6} + \left(12 a + 31\right)\cdot 41^{7} + \left(35 a + 30\right)\cdot 41^{8} + \left(a + 9\right)\cdot 41^{9} + \left(36 a + 17\right)\cdot 41^{10} + \left(14 a + 36\right)\cdot 41^{11} + \left(39 a + 29\right)\cdot 41^{12} +O\left(41^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,4)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,5,4,2)(3,6)$ | $1$ |
| $8$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
