Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(921\)\(\medspace = 3 \cdot 307 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.282747.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.921.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2763.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 2x^{4} + x^{3} + 2x^{2} - 3x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 41 + 40\cdot 73 + 49\cdot 73^{2} + 16\cdot 73^{3} + 41\cdot 73^{4} + 68\cdot 73^{5} + 40\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 64 a + 44 + \left(7 a + 55\right)\cdot 73 + \left(14 a + 26\right)\cdot 73^{2} + \left(53 a + 62\right)\cdot 73^{3} + \left(9 a + 13\right)\cdot 73^{4} + \left(52 a + 59\right)\cdot 73^{5} + \left(37 a + 26\right)\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 30 + \left(65 a + 17\right)\cdot 73 + \left(58 a + 46\right)\cdot 73^{2} + \left(19 a + 10\right)\cdot 73^{3} + \left(63 a + 59\right)\cdot 73^{4} + \left(20 a + 13\right)\cdot 73^{5} + \left(35 a + 46\right)\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 32\cdot 73 + 23\cdot 73^{2} + 56\cdot 73^{3} + 31\cdot 73^{4} + 4\cdot 73^{5} + 32\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 64 a + 57 + \left(7 a + 57\right)\cdot 73 + \left(14 a + 11\right)\cdot 73^{2} + \left(53 a + 11\right)\cdot 73^{3} + \left(9 a + 10\right)\cdot 73^{4} + \left(52 a + 13\right)\cdot 73^{5} + \left(37 a + 58\right)\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 17 + \left(65 a + 15\right)\cdot 73 + \left(58 a + 61\right)\cdot 73^{2} + \left(19 a + 61\right)\cdot 73^{3} + \left(63 a + 62\right)\cdot 73^{4} + \left(20 a + 59\right)\cdot 73^{5} + \left(35 a + 14\right)\cdot 73^{6} +O(73^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $6$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,5)(3,6,4)$ | $0$ |
| $6$ | $4$ | $(1,2,4,3)$ | $1$ |
| $6$ | $4$ | $(1,6,4,5)(2,3)$ | $-1$ |
| $8$ | $6$ | $(1,3,6,4,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.