Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(936\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.292032.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.104.2t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.104.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + x^{4} - 4x^{3} + x^{2} - 2x + 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 8 + \left(14 a + 12\right)\cdot 19 + 15\cdot 19^{2} + \left(15 a + 16\right)\cdot 19^{3} + \left(11 a + 15\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 13\cdot 19 + 6\cdot 19^{2} + 11\cdot 19^{4} +O(19^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 12 a + 15 + 4 a\cdot 19 + \left(18 a + 2\right)\cdot 19^{2} + \left(3 a + 12\right)\cdot 19^{3} + \left(7 a + 12\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 5\cdot 19 + 19^{2} + 9\cdot 19^{3} + 9\cdot 19^{4} +O(19^{5})\)
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| $r_{ 5 }$ | $=$ |
\( a + 12 + \left(8 a + 8\right)\cdot 19 + \left(14 a + 12\right)\cdot 19^{2} + \left(14 a + 18\right)\cdot 19^{3} + \left(9 a + 15\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a + 13 + \left(10 a + 15\right)\cdot 19 + \left(4 a + 18\right)\cdot 19^{2} + \left(4 a + 18\right)\cdot 19^{3} + \left(9 a + 10\right)\cdot 19^{4} +O(19^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-2$ | |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ | ✓ |
| $3$ | $2$ | $(1,3)(5,6)$ | $0$ | |
| $2$ | $3$ | $(1,4,3)(2,6,5)$ | $-1$ | |
| $2$ | $6$ | $(1,6,4,5,3,2)$ | $1$ |