Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1755\)\(\medspace = 3^{3} \cdot 5 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.9240075.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.1755.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 13\cdot 19 + 5\cdot 19^{2} + 11\cdot 19^{3} + 12\cdot 19^{4} + 13\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 5\cdot 19 + 13\cdot 19^{2} + 7\cdot 19^{3} + 6\cdot 19^{4} + 5\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 10 + \left(3 a + 11\right)\cdot 19 + \left(10 a + 18\right)\cdot 19^{2} + \left(16 a + 9\right)\cdot 19^{3} + \left(8 a + 6\right)\cdot 19^{4} + \left(10 a + 2\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 11 + \left(3 a + 2\right)\cdot 19 + \left(10 a + 13\right)\cdot 19^{2} + \left(16 a + 2\right)\cdot 19^{3} + \left(8 a + 1\right)\cdot 19^{4} + \left(10 a + 15\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 9 + \left(15 a + 16\right)\cdot 19 + \left(8 a + 5\right)\cdot 19^{2} + \left(2 a + 16\right)\cdot 19^{3} + \left(10 a + 17\right)\cdot 19^{4} + \left(8 a + 3\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 10 + \left(15 a + 7\right)\cdot 19 + 8 a\cdot 19^{2} + \left(2 a + 9\right)\cdot 19^{3} + \left(10 a + 12\right)\cdot 19^{4} + \left(8 a + 16\right)\cdot 19^{5} +O(19^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,4,6)$ | $-1$ |
| $2$ | $6$ | $(1,4,3,2,5,6)$ | $1$ |