Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3120\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.9734400.1 |
| 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
| $f(x)$ | $=$ |
\( x^{6} + 3x^{4} - 9x^{2} + 25 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 22 + \left(11 a + 5\right)\cdot 29 + \left(18 a + 17\right)\cdot 29^{2} + 12 a\cdot 29^{3} + \left(24 a + 5\right)\cdot 29^{4} + \left(19 a + 13\right)\cdot 29^{5} + \left(7 a + 11\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a + 10 + \left(17 a + 19\right)\cdot 29 + \left(10 a + 10\right)\cdot 29^{2} + \left(16 a + 16\right)\cdot 29^{3} + \left(4 a + 27\right)\cdot 29^{4} + 9 a\cdot 29^{5} + \left(21 a + 1\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 5\cdot 29 + 13\cdot 29^{2} + 13\cdot 29^{3} + 14\cdot 29^{4} + 2\cdot 29^{5} + 3\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a + 7 + \left(17 a + 23\right)\cdot 29 + \left(10 a + 11\right)\cdot 29^{2} + \left(16 a + 28\right)\cdot 29^{3} + \left(4 a + 23\right)\cdot 29^{4} + \left(9 a + 15\right)\cdot 29^{5} + \left(21 a + 17\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 19 + \left(11 a + 9\right)\cdot 29 + \left(18 a + 18\right)\cdot 29^{2} + \left(12 a + 12\right)\cdot 29^{3} + \left(24 a + 1\right)\cdot 29^{4} + \left(19 a + 28\right)\cdot 29^{5} + \left(7 a + 27\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 + 23\cdot 29 + 15\cdot 29^{2} + 15\cdot 29^{3} + 14\cdot 29^{4} + 26\cdot 29^{5} + 25\cdot 29^{6} +O(29^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ | |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ | |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ | ✓ |
| $2$ | $3$ | $(1,3,2)(4,6,5)$ | $-1$ | |
| $2$ | $6$ | $(1,6,2,4,3,5)$ | $1$ |