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 6.2.193600000.1 |
| 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 $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 31 + 22\cdot 37 + 21\cdot 37^{2} + 25\cdot 37^{3} + 3\cdot 37^{4} + 10\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 a + 29 + \left(23 a + 32\right)\cdot 37 + \left(21 a + 5\right)\cdot 37^{2} + \left(17 a + 6\right)\cdot 37^{3} + \left(34 a + 7\right)\cdot 37^{4} + \left(14 a + 6\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 8 + \left(31 a + 26\right)\cdot 37 + \left(24 a + 10\right)\cdot 37^{2} + \left(34 a + 4\right)\cdot 37^{3} + \left(33 a + 3\right)\cdot 37^{4} + \left(a + 8\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 25 + \left(13 a + 18\right)\cdot 37 + \left(15 a + 31\right)\cdot 37^{2} + \left(19 a + 17\right)\cdot 37^{3} + \left(2 a + 16\right)\cdot 37^{4} + \left(22 a + 31\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 + 22\cdot 37 + 36\cdot 37^{2} + 12\cdot 37^{3} + 13\cdot 37^{4} + 36\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 a + 35 + \left(5 a + 24\right)\cdot 37 + \left(12 a + 4\right)\cdot 37^{2} + \left(2 a + 7\right)\cdot 37^{3} + \left(3 a + 30\right)\cdot 37^{4} + \left(35 a + 18\right)\cdot 37^{5} +O(37^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-2$ |
| $3$ | $2$ | $(2,4)(3,6)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,6,3)(2,5,4)$ | $-1$ |
| $2$ | $6$ | $(1,2,6,5,3,4)$ | $1$ |