Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1323\)\(\medspace = 3^{3} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.12252303.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.1323.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 5 + a\cdot 11 + \left(8 a + 1\right)\cdot 11^{2} + \left(9 a + 8\right)\cdot 11^{3} + a\cdot 11^{4} + \left(9 a + 10\right)\cdot 11^{5} + \left(5 a + 3\right)\cdot 11^{6} +O(11^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 2\cdot 11 + 3\cdot 11^{3} + 10\cdot 11^{4} + 10\cdot 11^{5} + 10\cdot 11^{6} +O(11^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 10 + \left(9 a + 1\right)\cdot 11 + \left(2 a + 10\right)\cdot 11^{2} + \left(a + 5\right)\cdot 11^{3} + \left(9 a + 9\right)\cdot 11^{4} + a\cdot 11^{5} + \left(5 a + 7\right)\cdot 11^{6} +O(11^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 6 + \left(9 a + 10\right)\cdot 11 + \left(2 a + 9\right)\cdot 11^{2} + \left(a + 2\right)\cdot 11^{3} + \left(9 a + 10\right)\cdot 11^{4} + a\cdot 11^{5} + \left(5 a + 7\right)\cdot 11^{6} +O(11^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 + 8\cdot 11 + 10\cdot 11^{2} + 7\cdot 11^{3} +O(11^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 1 + \left(a + 9\right)\cdot 11 + 8 a\cdot 11^{2} + \left(9 a + 5\right)\cdot 11^{3} + \left(a + 1\right)\cdot 11^{4} + \left(9 a + 10\right)\cdot 11^{5} + \left(5 a + 3\right)\cdot 11^{6} +O(11^{7})\)
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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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,5,4,3,2)$ | $1$ |