Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2900\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.6728000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.116.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.116.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} + 6x^{3} + 17x^{2} + 16x + 16 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 11 + \left(12 a + 10\right)\cdot 23 + \left(15 a + 18\right)\cdot 23^{2} + \left(12 a + 14\right)\cdot 23^{3} + \left(9 a + 13\right)\cdot 23^{4} + \left(20 a + 13\right)\cdot 23^{5} + \left(9 a + 20\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 17 + 16\cdot 23 + 16\cdot 23^{2} + \left(20 a + 16\right)\cdot 23^{3} + \left(11 a + 18\right)\cdot 23^{4} + \left(22 a + 8\right)\cdot 23^{5} + \left(9 a + 1\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 + 10\cdot 23 + 23^{2} + 11\cdot 23^{3} + 17\cdot 23^{4} + 12\cdot 23^{5} + 4\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a + 6 + \left(10 a + 3\right)\cdot 23 + \left(7 a + 14\right)\cdot 23^{2} + \left(10 a + 1\right)\cdot 23^{3} + \left(13 a + 20\right)\cdot 23^{4} + \left(2 a + 21\right)\cdot 23^{5} + \left(13 a + 19\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 6 + \left(22 a + 11\right)\cdot 23 + \left(22 a + 16\right)\cdot 23^{2} + \left(2 a + 10\right)\cdot 23^{3} + \left(11 a + 22\right)\cdot 23^{4} + 18\cdot 23^{5} + \left(13 a + 21\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 + 16\cdot 23 + 23^{2} + 14\cdot 23^{3} + 22\cdot 23^{4} + 15\cdot 23^{5} +O(23^{7})\)
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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 value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-2$ | |
| $3$ | $2$ | $(2,3)(4,6)$ | $0$ | |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ | ✓ |
| $2$ | $3$ | $(1,6,4)(2,5,3)$ | $-1$ | |
| $2$ | $6$ | $(1,2,6,5,4,3)$ | $1$ |