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^{3} + x + 35 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 a^{2} + 14 a + 18 + \left(29 a^{2} + a + 24\right)\cdot 41 + \left(11 a^{2} + 34 a + 16\right)\cdot 41^{2} + \left(25 a^{2} + 37 a + 40\right)\cdot 41^{3} + \left(8 a^{2} + 23 a + 15\right)\cdot 41^{4} + \left(21 a^{2} + 36 a + 5\right)\cdot 41^{5} + \left(30 a^{2} + 27 a + 26\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 a^{2} + 28 a + 23 + \left(13 a^{2} + 16 a + 14\right)\cdot 41 + \left(10 a^{2} + 5 a + 24\right)\cdot 41^{2} + \left(26 a^{2} + 15 a + 16\right)\cdot 41^{3} + \left(3 a^{2} + 40 a + 29\right)\cdot 41^{4} + \left(10 a^{2} + 30 a + 29\right)\cdot 41^{5} + \left(24 a^{2} + 40 a + 36\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 a^{2} + 22 a + 27 + \left(24 a^{2} + 30 a + 35\right)\cdot 41 + \left(13 a^{2} + 17 a + 12\right)\cdot 41^{2} + \left(28 a^{2} + 19 a + 4\right)\cdot 41^{3} + \left(19 a^{2} + 32 a + 40\right)\cdot 41^{4} + \left(11 a^{2} + 6 a + 16\right)\cdot 41^{5} + \left(11 a^{2} + 33 a + 14\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 32\cdot 41 + 36\cdot 41^{2} + 40\cdot 41^{3} + 13\cdot 41^{4} + 14\cdot 41^{5} + 3\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 a^{2} + 32 a + 2 + \left(2 a^{2} + 34 a + 7\right)\cdot 41 + \left(17 a^{2} + 17 a + 15\right)\cdot 41^{2} + \left(27 a^{2} + 6 a + 17\right)\cdot 41^{3} + \left(17 a^{2} + 9 a + 11\right)\cdot 41^{4} + \left(19 a^{2} + 3 a + 22\right)\cdot 41^{5} + \left(5 a^{2} + 8 a + 10\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 + 8\cdot 41 + 31\cdot 41^{2} + 12\cdot 41^{4} + 28\cdot 41^{5} + 35\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 20 + 10\cdot 41 + 17\cdot 41^{2} + 13\cdot 41^{3} + 26\cdot 41^{4} + 37\cdot 41^{5} + 4\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 23 a^{2} + 33 a + 3 + \left(12 a^{2} + 7 a + 13\right)\cdot 41 + 36\cdot 41^{2} + \left(22 a^{2} + 21 a + 10\right)\cdot 41^{3} + \left(16 a^{2} + 40 a + 21\right)\cdot 41^{4} + \left(30 a^{2} + 4 a + 11\right)\cdot 41^{5} + \left(12 a^{2} + 36 a + 14\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 34 a^{2} + 35 a + 24 + \left(39 a^{2} + 31 a + 17\right)\cdot 41 + \left(28 a^{2} + 6 a + 14\right)\cdot 41^{2} + \left(34 a^{2} + 23 a + 19\right)\cdot 41^{3} + \left(15 a^{2} + 17 a + 34\right)\cdot 41^{4} + \left(30 a^{2} + 40 a + 38\right)\cdot 41^{5} + \left(38 a^{2} + 17 a + 17\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,7,2)(3,9,6)(4,5,8)$ |
| $(1,8,9)(4,7,6)$ |
| $(2,3,5)(4,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $1$ |
$3$ |
$(1,9,8)(2,3,5)(4,7,6)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $1$ |
$3$ |
$(1,8,9)(2,5,3)(4,6,7)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $3$ |
$3$ |
$(1,7,2)(3,9,6)(4,5,8)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,2,7)(3,6,9)(4,8,5)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,8,9)(4,7,6)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,9,8)(4,6,7)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,4,2)(3,9,7)(5,8,6)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,2,4)(3,7,9)(5,6,8)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,5,4)(2,7,9)(3,6,8)$ |
$0$ |
$0$ |
| $3$ |
$3$ |
$(1,4,5)(2,9,7)(3,8,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
