Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(2737\)\(\medspace = 7 \cdot 17 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.62951.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.2737.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.325703.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} - x^{3} + 4x^{2} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 23\cdot 61 + 30\cdot 61^{2} + 47\cdot 61^{3} + 11\cdot 61^{4} + 26\cdot 61^{5} + 38\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 a + 54 + \left(59 a + 34\right)\cdot 61 + \left(56 a + 7\right)\cdot 61^{2} + \left(53 a + 25\right)\cdot 61^{3} + \left(53 a + 56\right)\cdot 61^{4} + \left(2 a + 59\right)\cdot 61^{5} + \left(59 a + 18\right)\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 34 + \left(a + 53\right)\cdot 61 + \left(4 a + 4\right)\cdot 61^{2} + \left(7 a + 22\right)\cdot 61^{3} + \left(7 a + 56\right)\cdot 61^{4} + \left(58 a + 8\right)\cdot 61^{5} + \left(a + 14\right)\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 59 a + 26 + \left(9 a + 50\right)\cdot 61 + \left(43 a + 44\right)\cdot 61^{2} + \left(30 a + 60\right)\cdot 61^{3} + \left(34 a + 15\right)\cdot 61^{4} + \left(54 a + 33\right)\cdot 61^{5} + \left(23 a + 5\right)\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 24 + \left(51 a + 1\right)\cdot 61 + \left(17 a + 17\right)\cdot 61^{2} + \left(30 a + 48\right)\cdot 61^{3} + \left(26 a + 19\right)\cdot 61^{4} + \left(6 a + 53\right)\cdot 61^{5} + \left(37 a + 35\right)\cdot 61^{6} +O(61^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 28 + 19\cdot 61 + 17\cdot 61^{2} + 40\cdot 61^{3} + 22\cdot 61^{4} + 61^{5} + 9\cdot 61^{6} +O(61^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,4)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,4,6,2)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,5,4,3)$ | $-1$ |
| $8$ | $6$ | $(1,4,5,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.