Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1456\)\(\medspace = 2^{4} \cdot 7 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.14839552.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.364.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 17 a + 32 + \left(23 a + 14\right)\cdot 37 + \left(11 a + 34\right)\cdot 37^{2} + \left(16 a + 9\right)\cdot 37^{3} + \left(10 a + 15\right)\cdot 37^{4} + \left(21 a + 27\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 + 36\cdot 37 + 16\cdot 37^{2} + 20\cdot 37^{3} + 7\cdot 37^{4} + 36\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + 11 + \left(23 a + 19\right)\cdot 37 + \left(11 a + 16\right)\cdot 37^{2} + \left(16 a + 10\right)\cdot 37^{3} + \left(10 a + 33\right)\cdot 37^{4} + \left(21 a + 8\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 5 + \left(13 a + 22\right)\cdot 37 + \left(25 a + 2\right)\cdot 37^{2} + \left(20 a + 27\right)\cdot 37^{3} + \left(26 a + 21\right)\cdot 37^{4} + \left(15 a + 9\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 + 20\cdot 37^{2} + 16\cdot 37^{3} + 29\cdot 37^{4} +O(37^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a + 26 + \left(13 a + 17\right)\cdot 37 + \left(25 a + 20\right)\cdot 37^{2} + \left(20 a + 26\right)\cdot 37^{3} + \left(26 a + 3\right)\cdot 37^{4} + \left(15 a + 28\right)\cdot 37^{5} +O(37^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $0$ |
| $2$ | $3$ | $(1,5,6)(2,3,4)$ | $-1$ |
| $2$ | $6$ | $(1,3,5,4,6,2)$ | $1$ |