Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(14764742975\)\(\medspace = 5^{2} \cdot 839^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.2952948595.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | odd |
| Determinant: | 1.839.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.2952948595.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 9x^{4} - 19x^{3} + 27x^{2} - 44x + 240 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 18 + 9\cdot 19 + 11\cdot 19^{2} + 5\cdot 19^{3} + 4\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 7 + \left(5 a + 9\right)\cdot 19 + \left(8 a + 14\right)\cdot 19^{2} + \left(14 a + 6\right)\cdot 19^{3} + \left(8 a + 12\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 7\cdot 19 + 6\cdot 19^{2} + 18\cdot 19^{3} + 18\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 14 + \left(7 a + 6\right)\cdot 19 + \left(9 a + 12\right)\cdot 19^{2} + \left(a + 10\right)\cdot 19^{3} + \left(7 a + 4\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + 1 + \left(13 a + 2\right)\cdot 19 + \left(10 a + 17\right)\cdot 19^{2} + \left(4 a + 12\right)\cdot 19^{3} + \left(10 a + 6\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 7 + \left(11 a + 2\right)\cdot 19 + \left(9 a + 14\right)\cdot 19^{2} + \left(17 a + 2\right)\cdot 19^{3} + \left(11 a + 10\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ | |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ | ✓ |
| $6$ | $2$ | $(2,3)$ | $0$ | |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ | |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ | |
| $4$ | $3$ | $(1,4,6)$ | $-2$ | |
| $18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ | |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $1$ | |
| $12$ | $6$ | $(1,4,6)(2,3)$ | $0$ |