Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1815\)\(\medspace = 3 \cdot 5 \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.108709425.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.1815.1 |
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:
| $r_{ 1 }$ | $=$ |
\( 14\cdot 29 + 27\cdot 29^{2} + 3\cdot 29^{3} + 19\cdot 29^{4} + 27\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 28\cdot 29 + 11\cdot 29^{2} + 3\cdot 29^{3} + 21\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 4 a\cdot 29 + \left(24 a + 15\right)\cdot 29^{2} + \left(8 a + 2\right)\cdot 29^{3} + \left(12 a + 22\right)\cdot 29^{4} + \left(8 a + 14\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 a + 1 + \left(24 a + 15\right)\cdot 29 + \left(4 a + 15\right)\cdot 29^{2} + \left(20 a + 22\right)\cdot 29^{3} + \left(16 a + 16\right)\cdot 29^{4} + \left(20 a + 15\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 1 + \left(6 a + 2\right)\cdot 29 + \left(20 a + 19\right)\cdot 29^{2} + \left(6 a + 20\right)\cdot 29^{3} + \left(6 a + 16\right)\cdot 29^{4} + \left(21 a + 26\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a + 26 + \left(22 a + 27\right)\cdot 29 + \left(8 a + 26\right)\cdot 29^{2} + \left(22 a + 4\right)\cdot 29^{3} + \left(22 a + 12\right)\cdot 29^{4} + \left(7 a + 10\right)\cdot 29^{5} +O(29^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,2,4,5)$ | $1$ |