Basic invariants
Dimension: | $3$ |
Group: | $S_4\times C_2$ |
Conductor: | \(437\)\(\medspace = 19 \cdot 23 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.10051.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4\times C_2$ |
Parity: | even |
Determinant: | 1.437.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.8303.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 2x^{4} - 2x^{3} + 2x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 3\cdot 11 + 10\cdot 11^{2} + 4\cdot 11^{3} + 6\cdot 11^{4} + 5\cdot 11^{5} + 9\cdot 11^{6} + 5\cdot 11^{7} +O(11^{8})\) |
$r_{ 2 }$ | $=$ | \( 10 + 4\cdot 11 + 2\cdot 11^{3} + 11^{4} + 6\cdot 11^{5} + 7\cdot 11^{7} +O(11^{8})\) |
$r_{ 3 }$ | $=$ | \( 7 a + 6 + \left(7 a + 4\right)\cdot 11 + \left(5 a + 6\right)\cdot 11^{2} + \left(4 a + 3\right)\cdot 11^{3} + 10 a\cdot 11^{4} + \left(8 a + 9\right)\cdot 11^{5} + \left(2 a + 3\right)\cdot 11^{6} + \left(7 a + 3\right)\cdot 11^{7} +O(11^{8})\) |
$r_{ 4 }$ | $=$ | \( 3 a + \left(9 a + 1\right)\cdot 11 + \left(3 a + 5\right)\cdot 11^{2} + \left(8 a + 10\right)\cdot 11^{3} + \left(2 a + 8\right)\cdot 11^{4} + \left(4 a + 3\right)\cdot 11^{5} + \left(5 a + 3\right)\cdot 11^{6} + \left(6 a + 5\right)\cdot 11^{7} +O(11^{8})\) |
$r_{ 5 }$ | $=$ | \( 4 a + 1 + \left(3 a + 6\right)\cdot 11 + \left(5 a + 10\right)\cdot 11^{2} + \left(6 a + 4\right)\cdot 11^{3} + 4\cdot 11^{4} + \left(2 a + 1\right)\cdot 11^{5} + \left(8 a + 6\right)\cdot 11^{6} + \left(3 a + 7\right)\cdot 11^{7} +O(11^{8})\) |
$r_{ 6 }$ | $=$ | \( 8 a + 1 + \left(a + 2\right)\cdot 11 + 7 a\cdot 11^{2} + \left(2 a + 7\right)\cdot 11^{3} + 8 a\cdot 11^{4} + \left(6 a + 7\right)\cdot 11^{5} + \left(5 a + 9\right)\cdot 11^{6} + \left(4 a + 3\right)\cdot 11^{7} +O(11^{8})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
$3$ | $2$ | $(1,2)$ | $1$ |
$3$ | $2$ | $(1,2)(5,6)$ | $-1$ |
$6$ | $2$ | $(3,5)(4,6)$ | $1$ |
$6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
$8$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ |
$6$ | $4$ | $(1,6,2,5)$ | $1$ |
$6$ | $4$ | $(1,2)(3,6,4,5)$ | $-1$ |
$8$ | $6$ | $(1,6,4,2,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.