Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3744\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.9345024.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.104.1 |
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:
| $r_{ 1 }$ | $=$ |
\( 24 a + 19 + \left(20 a + 6\right)\cdot 29 + \left(20 a + 16\right)\cdot 29^{2} + \left(7 a + 22\right)\cdot 29^{3} + \left(10 a + 23\right)\cdot 29^{4} + \left(27 a + 17\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 2\cdot 29 + 8\cdot 29^{2} + 27\cdot 29^{3} + 2\cdot 29^{4} + 6\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 16 + \left(7 a + 17\right)\cdot 29 + \left(27 a + 18\right)\cdot 29^{2} + \left(19 a + 22\right)\cdot 29^{3} + \left(15 a + 12\right)\cdot 29^{4} + \left(28 a + 20\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 23 + \left(8 a + 28\right)\cdot 29 + \left(8 a + 11\right)\cdot 29^{2} + \left(21 a + 11\right)\cdot 29^{3} + \left(18 a + 9\right)\cdot 29^{4} + \left(a + 28\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 19 + \left(21 a + 8\right)\cdot 29 + \left(a + 2\right)\cdot 29^{2} + \left(9 a + 8\right)\cdot 29^{3} + \left(13 a + 13\right)\cdot 29^{4} + 2\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 + 22\cdot 29 + 24\cdot 29^{3} + 24\cdot 29^{4} + 11\cdot 29^{5} +O(29^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,4)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,6,4)(2,5,3)$ | $-1$ |
| $2$ | $6$ | $(1,5,6,3,4,2)$ | $1$ |