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 6.2.37850112.1 |
| 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 $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 25 + 24\cdot 47 + 31\cdot 47^{2} + 46\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 24 + \left(2 a + 13\right)\cdot 47 + \left(21 a + 27\right)\cdot 47^{2} + \left(30 a + 32\right)\cdot 47^{3} + \left(3 a + 9\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 a + 26 + \left(44 a + 16\right)\cdot 47 + \left(25 a + 20\right)\cdot 47^{2} + \left(16 a + 25\right)\cdot 47^{3} + \left(43 a + 33\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 a + 43 + \left(8 a + 21\right)\cdot 47 + \left(30 a + 5\right)\cdot 47^{2} + \left(46 a + 15\right)\cdot 47^{3} + \left(18 a + 28\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 27 + 38 a\cdot 47 + \left(16 a + 10\right)\cdot 47^{2} + 31\cdot 47^{3} + \left(28 a + 19\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 45 + 16\cdot 47 + 46\cdot 47^{2} + 35\cdot 47^{3} + 3\cdot 47^{4} +O(47^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,5,4)(2,3,6)$ | $-1$ |
| $2$ | $6$ | $(1,3,5,6,4,2)$ | $1$ |