Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.24200000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.2200.1 |
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 $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
![Copy content]()
![Toggle raw display]()
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 36\cdot 41 + 29\cdot 41^{2} + 10\cdot 41^{3} + 5\cdot 41^{4} + 33\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 4\cdot 41 + 11\cdot 41^{2} + 30\cdot 41^{3} + 35\cdot 41^{4} + 7\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 10 + \left(20 a + 3\right)\cdot 41 + \left(4 a + 40\right)\cdot 41^{2} + \left(35 a + 20\right)\cdot 41^{3} + \left(32 a + 33\right)\cdot 41^{4} + 7\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 a + 32 + \left(20 a + 37\right)\cdot 41 + 36 a\cdot 41^{2} + \left(5 a + 20\right)\cdot 41^{3} + \left(8 a + 7\right)\cdot 41^{4} + \left(40 a + 33\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 a + 16 + \left(20 a + 20\right)\cdot 41 + \left(36 a + 33\right)\cdot 41^{2} + \left(5 a + 39\right)\cdot 41^{3} + \left(8 a + 14\right)\cdot 41^{4} + \left(40 a + 18\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 26 + \left(20 a + 20\right)\cdot 41 + \left(4 a + 7\right)\cdot 41^{2} + \left(35 a + 1\right)\cdot 41^{3} + \left(32 a + 26\right)\cdot 41^{4} + 22\cdot 41^{5} +O(41^{6})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $3$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,3,5)(2,4,6)$ | $-1$ |
| $2$ | $6$ | $(1,4,5,2,3,6)$ | $1$ |