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