Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(888\)\(\medspace = 2^{3} \cdot 3 \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.18925056.1 |
| 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 $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 6\cdot 29^{2} + 2\cdot 29^{3} + 24\cdot 29^{4} + 16\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 20 + \left(3 a + 26\right)\cdot 29 + \left(16 a + 1\right)\cdot 29^{2} + \left(23 a + 6\right)\cdot 29^{3} + \left(12 a + 11\right)\cdot 29^{4} + \left(27 a + 16\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 25\cdot 29 + 27\cdot 29^{2} + 21\cdot 29^{3} + 29^{4} + 28\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 3 + \left(22 a + 20\right)\cdot 29 + \left(3 a + 16\right)\cdot 29^{2} + \left(10 a + 23\right)\cdot 29^{3} + \left(5 a + 19\right)\cdot 29^{4} + \left(25 a + 12\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 27 + \left(25 a + 1\right)\cdot 29 + \left(12 a + 21\right)\cdot 29^{2} + \left(5 a + 20\right)\cdot 29^{3} + \left(16 a + 22\right)\cdot 29^{4} + \left(a + 24\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 a + 13 + \left(6 a + 12\right)\cdot 29 + \left(25 a + 13\right)\cdot 29^{2} + \left(18 a + 12\right)\cdot 29^{3} + \left(23 a + 7\right)\cdot 29^{4} + \left(3 a + 17\right)\cdot 29^{5} +O(29^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,5)(3,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,5,3,2,4)$ | $1$ |