Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{3} + x + 35 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a^{2} + 26 a + 12 + \left(7 a^{2} + 30 a + 31\right)\cdot 41 + \left(a^{2} + 21 a + 5\right)\cdot 41^{2} + \left(30 a^{2} + 5 a + 17\right)\cdot 41^{3} + \left(36 a^{2} + 12 a + 21\right)\cdot 41^{4} + \left(21 a^{2} + 2 a + 37\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a^{2} + 31 a + 22 + \left(19 a^{2} + 20 a + 11\right)\cdot 41 + \left(2 a^{2} + 3 a + 21\right)\cdot 41^{2} + \left(27 a^{2} + 40 a + 26\right)\cdot 41^{3} + \left(a^{2} + 34 a + 35\right)\cdot 41^{4} + \left(3 a^{2} + 16 a + 23\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 8\cdot 41 + 3\cdot 41^{2} + 38\cdot 41^{4} + 16\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 13\cdot 41 + 20\cdot 41^{2} + 7\cdot 41^{3} + 2\cdot 41^{4} + 3\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a^{2} + 33 a + 12 + \left(5 a^{2} + 8 a + 16\right)\cdot 41 + \left(12 a^{2} + 31 a + 40\right)\cdot 41^{2} + \left(17 a^{2} + 39 a + 35\right)\cdot 41^{3} + \left(7 a^{2} + 36 a + 1\right)\cdot 41^{4} + \left(35 a^{2} + 20 a + 19\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a^{2} + 31 a + 39 + \left(19 a^{2} + 38\right)\cdot 41 + \left(11 a^{2} + 29 a + 40\right)\cdot 41^{2} + \left(40 a^{2} + 22 a + 7\right)\cdot 41^{3} + \left(39 a^{2} + 26 a + 20\right)\cdot 41^{4} + \left(18 a^{2} + 18 a + 34\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 a^{2} + 20 a + 3 + \left(2 a^{2} + 19 a + 28\right)\cdot 41 + \left(27 a^{2} + 8 a + 37\right)\cdot 41^{2} + \left(14 a^{2} + 19 a + 31\right)\cdot 41^{3} + \left(40 a^{2} + 20 a + 6\right)\cdot 41^{4} + \left(18 a^{2} + 5 a + 7\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 23 + 24\cdot 41 + 25\cdot 41^{2} + 16\cdot 41^{3} + 29\cdot 41^{4} + 9\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 39 a^{2} + 23 a + 10 + \left(28 a^{2} + a + 32\right)\cdot 41 + \left(27 a^{2} + 29 a + 9\right)\cdot 41^{2} + \left(34 a^{2} + 36 a + 20\right)\cdot 41^{3} + \left(37 a^{2} + 32 a + 8\right)\cdot 41^{4} + \left(24 a^{2} + 17 a + 12\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5,9)(2,7,6)$ |
| $(2,6,7)(3,8,4)$ |
| $(1,7,3,5,2,4,9,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $3$ | $(1,5,9)(2,6,7)(3,4,8)$ | $-3 \zeta_{3} - 3$ |
| $1$ | $3$ | $(1,9,5)(2,7,6)(3,8,4)$ | $3 \zeta_{3}$ |
| $3$ | $3$ | $(1,5,9)(2,7,6)$ | $0$ |
| $3$ | $3$ | $(1,9,5)(2,6,7)$ | $0$ |
| $3$ | $9$ | $(1,7,3,5,2,4,9,6,8)$ | $0$ |
| $3$ | $9$ | $(1,3,2,9,8,7,5,4,6)$ | $0$ |
| $3$ | $9$ | $(1,6,8,5,7,3,9,2,4)$ | $0$ |
| $3$ | $9$ | $(1,8,7,9,4,6,5,3,2)$ | $0$ |
| $3$ | $9$ | $(1,4,6,9,3,2,5,8,7)$ | $0$ |
| $3$ | $9$ | $(1,6,3,5,7,4,9,2,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
