Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3724\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.97077232.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.3724.2 |
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:
| $r_{ 1 }$ | $=$ |
\( 12 + 12\cdot 29 + 29^{2} + 3\cdot 29^{3} + 8\cdot 29^{4} + 28\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 16\cdot 29 + 27\cdot 29^{2} + 25\cdot 29^{3} + 20\cdot 29^{4} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 1 + \left(4 a + 20\right)\cdot 29 + \left(16 a + 5\right)\cdot 29^{2} + \left(4 a + 20\right)\cdot 29^{3} + 17\cdot 29^{4} + \left(24 a + 13\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 4 + \left(24 a + 25\right)\cdot 29 + \left(12 a + 23\right)\cdot 29^{2} + \left(24 a + 26\right)\cdot 29^{3} + \left(28 a + 13\right)\cdot 29^{4} + \left(4 a + 17\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + \left(24 a + 9\right)\cdot 29 + \left(12 a + 23\right)\cdot 29^{2} + \left(24 a + 8\right)\cdot 29^{3} + \left(28 a + 11\right)\cdot 29^{4} + \left(4 a + 15\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a + 26 + \left(4 a + 3\right)\cdot 29 + \left(16 a + 5\right)\cdot 29^{2} + \left(4 a + 2\right)\cdot 29^{3} + 15\cdot 29^{4} + \left(24 a + 11\right)\cdot 29^{5} +O(29^{6})\)
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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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $2$ | $3$ | $(1,5,6)(2,3,4)$ | $-1$ |
| $2$ | $6$ | $(1,4,5,2,6,3)$ | $1$ |