Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3332\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.1321164656.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.68.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.3332.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 3x^{4} - 8x^{3} + 135x^{2} - 214x + 155 \)
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The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 a + 17 + \left(3 a + 17\right)\cdot 41 + \left(39 a + 31\right)\cdot 41^{2} + \left(18 a + 21\right)\cdot 41^{3} + \left(34 a + 4\right)\cdot 41^{4} + \left(33 a + 24\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 40 + \left(37 a + 34\right)\cdot 41 + \left(a + 21\right)\cdot 41^{2} + \left(22 a + 39\right)\cdot 41^{3} + \left(6 a + 6\right)\cdot 41^{4} + \left(7 a + 9\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 29\cdot 41 + 28\cdot 41^{2} + 20\cdot 41^{3} + 29\cdot 41^{4} + 7\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 + 36\cdot 41 + 35\cdot 41^{2} + 15\cdot 41^{3} + 14\cdot 41^{4} + 10\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 a + 40 + \left(24 a + 17\right)\cdot 41 + \left(7 a + 3\right)\cdot 41^{2} + \left(18 a + 30\right)\cdot 41^{3} + \left(27 a + 1\right)\cdot 41^{4} + \left(24 a + 33\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a + 27 + \left(16 a + 27\right)\cdot 41 + \left(33 a + 1\right)\cdot 41^{2} + \left(22 a + 36\right)\cdot 41^{3} + \left(13 a + 24\right)\cdot 41^{4} + \left(16 a + 38\right)\cdot 41^{5} +O(41^{6})\)
|
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 value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-2$ | |
| $3$ | $2$ | $(1,2)(5,6)$ | $0$ | |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ | ✓ |
| $2$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ | |
| $2$ | $6$ | $(1,4,2,5,3,6)$ | $1$ |