Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3639\)\(\medspace = 3 \cdot 1213 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.39726963.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.3639.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 28 + \left(8 a + 19\right)\cdot 37 + \left(8 a + 8\right)\cdot 37^{2} + \left(8 a + 21\right)\cdot 37^{3} + \left(4 a + 3\right)\cdot 37^{4} + \left(24 a + 31\right)\cdot 37^{5} + \left(6 a + 13\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 22 + \left(8 a + 27\right)\cdot 37 + \left(8 a + 3\right)\cdot 37^{2} + \left(8 a + 28\right)\cdot 37^{3} + \left(4 a + 24\right)\cdot 37^{4} + \left(24 a + 24\right)\cdot 37^{5} + \left(6 a + 20\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 13\cdot 37 + 11\cdot 37^{2} + 23\cdot 37^{3} + 22\cdot 37^{4} + 34\cdot 37^{5} + 18\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 a + 9 + \left(28 a + 17\right)\cdot 37 + \left(28 a + 28\right)\cdot 37^{2} + \left(28 a + 15\right)\cdot 37^{3} + \left(32 a + 33\right)\cdot 37^{4} + \left(12 a + 5\right)\cdot 37^{5} + \left(30 a + 23\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 a + 15 + \left(28 a + 9\right)\cdot 37 + \left(28 a + 33\right)\cdot 37^{2} + \left(28 a + 8\right)\cdot 37^{3} + \left(32 a + 12\right)\cdot 37^{4} + \left(12 a + 12\right)\cdot 37^{5} + \left(30 a + 16\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 + 23\cdot 37 + 25\cdot 37^{2} + 13\cdot 37^{3} + 14\cdot 37^{4} + 2\cdot 37^{5} + 18\cdot 37^{6} +O(37^{7})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,5,4,3,2)$ | $1$ |