Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(84255125\)\(\medspace = 5^{3} \cdot 821^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.2766938305.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.5.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.2766938305.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} + 29x^{3} - 29x^{2} + 5 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 4 a + 12 + \left(13 a + 13\right)\cdot 19 + \left(12 a + 7\right)\cdot 19^{2} + \left(14 a + 2\right)\cdot 19^{3} + \left(16 a + 1\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 14 + 5\cdot 19 + \left(6 a + 17\right)\cdot 19^{2} + \left(10 a + 8\right)\cdot 19^{3} + \left(14 a + 14\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 16 + \left(5 a + 3\right)\cdot 19 + \left(6 a + 7\right)\cdot 19^{2} + \left(4 a + 4\right)\cdot 19^{3} + \left(2 a + 3\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 2\cdot 19 + 17\cdot 19^{2} + 15\cdot 19^{3} + 4\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 + 19 + 4\cdot 19^{2} + 12\cdot 19^{3} + 14\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 10 + \left(18 a + 10\right)\cdot 19 + \left(12 a + 3\right)\cdot 19^{2} + \left(8 a + 13\right)\cdot 19^{3} + \left(4 a + 18\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ | |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ | |
| $6$ | $2$ | $(1,3)$ | $-2$ | |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ | ✓ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ | |
| $4$ | $3$ | $(1,3,5)$ | $1$ | |
| $18$ | $4$ | $(1,4,3,2)(5,6)$ | $0$ | |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $0$ | |
| $12$ | $6$ | $(1,3)(2,4,6)$ | $1$ |