Basic invariants
Dimension: | $3$ |
Group: | $S_4\times C_2$ |
Conductor: | \(1472\)\(\medspace = 2^{6} \cdot 23 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.33856.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4\times C_2$ |
Parity: | even |
Determinant: | 1.92.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.1472.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{4} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 50 a + 8 + \left(30 a + 19\right)\cdot 53 + \left(24 a + 47\right)\cdot 53^{2} + \left(21 a + 12\right)\cdot 53^{3} + \left(39 a + 17\right)\cdot 53^{4} + 7\cdot 53^{5} + \left(13 a + 46\right)\cdot 53^{6} +O(53^{7})\) |
$r_{ 2 }$ | $=$ | \( 50 a + 4 + \left(30 a + 13\right)\cdot 53 + \left(24 a + 44\right)\cdot 53^{2} + \left(21 a + 31\right)\cdot 53^{3} + \left(39 a + 5\right)\cdot 53^{4} + 29\cdot 53^{5} + \left(13 a + 8\right)\cdot 53^{6} +O(53^{7})\) |
$r_{ 3 }$ | $=$ | \( 13 + 33\cdot 53 + 46\cdot 53^{2} + 34\cdot 53^{3} + 27\cdot 53^{4} + 50\cdot 53^{5} + 4\cdot 53^{6} +O(53^{7})\) |
$r_{ 4 }$ | $=$ | \( 3 a + 45 + \left(22 a + 33\right)\cdot 53 + \left(28 a + 5\right)\cdot 53^{2} + \left(31 a + 40\right)\cdot 53^{3} + \left(13 a + 35\right)\cdot 53^{4} + \left(52 a + 45\right)\cdot 53^{5} + \left(39 a + 6\right)\cdot 53^{6} +O(53^{7})\) |
$r_{ 5 }$ | $=$ | \( 3 a + 49 + \left(22 a + 39\right)\cdot 53 + \left(28 a + 8\right)\cdot 53^{2} + \left(31 a + 21\right)\cdot 53^{3} + \left(13 a + 47\right)\cdot 53^{4} + \left(52 a + 23\right)\cdot 53^{5} + \left(39 a + 44\right)\cdot 53^{6} +O(53^{7})\) |
$r_{ 6 }$ | $=$ | \( 40 + 19\cdot 53 + 6\cdot 53^{2} + 18\cdot 53^{3} + 25\cdot 53^{4} + 2\cdot 53^{5} + 48\cdot 53^{6} +O(53^{7})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
$3$ | $2$ | $(1,4)$ | $1$ |
$3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
$6$ | $2$ | $(2,3)(5,6)$ | $1$ |
$6$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
$8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
$6$ | $4$ | $(1,6,4,3)$ | $1$ |
$6$ | $4$ | $(1,6,4,3)(2,5)$ | $-1$ |
$8$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.