Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(780\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.1825200.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.195.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.780.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} + 11x^{3} - 4x^{2} + 5x + 25 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 4\cdot 19 + 10\cdot 19^{2} + 19^{3} + 5\cdot 19^{4} + 6\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 8 + \left(7 a + 1\right)\cdot 19 + \left(18 a + 1\right)\cdot 19^{2} + \left(a + 7\right)\cdot 19^{3} + \left(6 a + 8\right)\cdot 19^{4} + \left(13 a + 5\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 19 + 14\cdot 19^{2} + 2\cdot 19^{3} + 18\cdot 19^{4} + 18\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a + 17 + \left(17 a + 16\right)\cdot 19 + \left(15 a + 10\right)\cdot 19^{2} + \left(4 a + 4\right)\cdot 19^{3} + \left(14 a + 11\right)\cdot 19^{4} + \left(13 a + 16\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 12 + \left(a + 1\right)\cdot 19 + \left(3 a + 9\right)\cdot 19^{2} + \left(14 a + 12\right)\cdot 19^{3} + \left(4 a + 1\right)\cdot 19^{4} + \left(5 a + 16\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 5 + \left(11 a + 12\right)\cdot 19 + 11\cdot 19^{2} + \left(17 a + 9\right)\cdot 19^{3} + \left(12 a + 12\right)\cdot 19^{4} + \left(5 a + 12\right)\cdot 19^{5} +O(19^{6})\)
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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 value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-2$ | |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ | ✓ |
| $3$ | $2$ | $(1,4)(3,6)$ | $0$ | |
| $2$ | $3$ | $(1,5,4)(2,6,3)$ | $-1$ | |
| $2$ | $6$ | $(1,6,5,3,4,2)$ | $1$ |