Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(8956368\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \cdot 41^{2}\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.299055312.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.37.2t1.a.a |
| Projective image: | $S_3\wr C_2$ |
| Projective stem field: | Galois closure of 6.2.299055312.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 10x^{4} + 5x^{3} + 46x^{2} - x - 33 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 a + 34 + \left(25 a + 51\right)\cdot 73 + \left(47 a + 27\right)\cdot 73^{2} + \left(46 a + 18\right)\cdot 73^{3} + \left(39 a + 50\right)\cdot 73^{4} +O(73^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 60 a + 50 + \left(11 a + 60\right)\cdot 73 + \left(53 a + 36\right)\cdot 73^{2} + \left(6 a + 13\right)\cdot 73^{3} + \left(58 a + 10\right)\cdot 73^{4} +O(73^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 13 a + 11 + \left(61 a + 36\right)\cdot 73 + \left(19 a + 38\right)\cdot 73^{2} + \left(66 a + 53\right)\cdot 73^{3} + \left(14 a + 31\right)\cdot 73^{4} +O(73^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 50 a + 30 + \left(47 a + 31\right)\cdot 73 + \left(25 a + 71\right)\cdot 73^{2} + \left(26 a + 37\right)\cdot 73^{3} + \left(33 a + 49\right)\cdot 73^{4} +O(73^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 57 + 61\cdot 73 + 19\cdot 73^{2} + 67\cdot 73^{3} + 41\cdot 73^{4} +O(73^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 38 + 50\cdot 73 + 24\cdot 73^{2} + 28\cdot 73^{3} + 35\cdot 73^{4} +O(73^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(1,4)$ | $0$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ |
| $4$ | $3$ | $(2,3,5)$ | $-2$ |
| $18$ | $4$ | $(1,3,4,2)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,6,5)$ | $-1$ |
| $12$ | $6$ | $(1,4)(2,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.