Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3552\)\(\medspace = 2^{5} \cdot 3 \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.25233408.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.888.1 |
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 $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 27 + 16\cdot 41 + 29\cdot 41^{2} + 16\cdot 41^{3} + 6\cdot 41^{4} + 16\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 37 + \left(5 a + 25\right)\cdot 41 + \left(6 a + 29\right)\cdot 41^{2} + \left(34 a + 37\right)\cdot 41^{3} + \left(19 a + 1\right)\cdot 41^{4} + \left(17 a + 20\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a + 20 + \left(35 a + 33\right)\cdot 41 + \left(34 a + 1\right)\cdot 41^{2} + \left(6 a + 11\right)\cdot 41^{3} + \left(21 a + 27\right)\cdot 41^{4} + \left(23 a + 11\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 + 24\cdot 41 + 11\cdot 41^{2} + 24\cdot 41^{3} + 34\cdot 41^{4} + 24\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 a + 4 + \left(35 a + 15\right)\cdot 41 + \left(34 a + 11\right)\cdot 41^{2} + \left(6 a + 3\right)\cdot 41^{3} + \left(21 a + 39\right)\cdot 41^{4} + \left(23 a + 20\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a + 21 + \left(5 a + 7\right)\cdot 41 + \left(6 a + 39\right)\cdot 41^{2} + \left(34 a + 29\right)\cdot 41^{3} + \left(19 a + 13\right)\cdot 41^{4} + \left(17 a + 29\right)\cdot 41^{5} +O(41^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,6,5)(2,4,3)$ | $-1$ |
| $2$ | $6$ | $(1,2,6,4,5,3)$ | $1$ |