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