Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(740\)\(\medspace = 2^{2} \cdot 5 \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.109520.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.185.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.3700.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{3} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 53 + \left(32 a + 22\right)\cdot 59 + \left(39 a + 37\right)\cdot 59^{2} + \left(49 a + 58\right)\cdot 59^{3} + \left(42 a + 17\right)\cdot 59^{4} + \left(27 a + 13\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 56 + 44\cdot 59 + 33\cdot 59^{2} + 59^{3} + 53\cdot 59^{4} + 37\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 55 a + 57 + \left(26 a + 50\right)\cdot 59 + \left(19 a + 44\right)\cdot 59^{2} + \left(9 a + 9\right)\cdot 59^{3} + \left(16 a + 11\right)\cdot 59^{4} + \left(31 a + 57\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 a + 24 + 26 a\cdot 59 + \left(24 a + 12\right)\cdot 59^{2} + \left(28 a + 14\right)\cdot 59^{3} + \left(53 a + 23\right)\cdot 59^{4} + \left(39 a + 23\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 + 14\cdot 59 + 39\cdot 59^{2} + 15\cdot 59^{3} + 23\cdot 59^{4} + 35\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 8 + \left(32 a + 43\right)\cdot 59 + \left(34 a + 9\right)\cdot 59^{2} + \left(30 a + 18\right)\cdot 59^{3} + \left(5 a + 48\right)\cdot 59^{4} + \left(19 a + 9\right)\cdot 59^{5} +O(59^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $1$ |
| $6$ | $4$ | $(1,4,6,3)(2,5)$ | $-1$ |
| $8$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.