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 6.2.40040325.1 |
| 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 $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( a + 15 + \left(4 a + 28\right)\cdot 29 + \left(5 a + 22\right)\cdot 29^{2} + \left(19 a + 2\right)\cdot 29^{3} + \left(7 a + 9\right)\cdot 29^{4} + \left(15 a + 5\right)\cdot 29^{5} + \left(4 a + 7\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 10 + \left(16 a + 11\right)\cdot 29 + \left(23 a + 5\right)\cdot 29^{2} + \left(16 a + 5\right)\cdot 29^{3} + \left(2 a + 9\right)\cdot 29^{4} + 9\cdot 29^{5} + \left(15 a + 26\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 a + 20 + \left(24 a + 18\right)\cdot 29 + \left(23 a + 15\right)\cdot 29^{2} + \left(9 a + 6\right)\cdot 29^{3} + \left(21 a + 28\right)\cdot 29^{4} + \left(13 a + 15\right)\cdot 29^{5} + \left(24 a + 14\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 + 14\cdot 29 + 3\cdot 29^{2} + 16\cdot 29^{3} + 14\cdot 29^{4} + 12\cdot 29^{5} + 17\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 a + 20 + \left(12 a + 2\right)\cdot 29 + \left(5 a + 20\right)\cdot 29^{2} + \left(12 a + 7\right)\cdot 29^{3} + \left(26 a + 5\right)\cdot 29^{4} + \left(28 a + 7\right)\cdot 29^{5} + \left(13 a + 14\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 + 10\cdot 29 + 19\cdot 29^{2} + 19\cdot 29^{3} + 20\cdot 29^{4} + 7\cdot 29^{5} + 7\cdot 29^{6} +O(29^{7})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,6,3)(2,4,5)$ | $-1$ |
| $2$ | $6$ | $(1,4,3,2,6,5)$ | $1$ |