Basic invariants
Dimension: | $3$ |
Group: | $S_4\times C_2$ |
Conductor: | \(749\)\(\medspace = 7 \cdot 107 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.80143.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4\times C_2$ |
Parity: | even |
Determinant: | 1.749.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.5243.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + x^{4} + x^{3} + x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 10 a + 26 + \left(15 a + 26\right)\cdot 29 + \left(3 a + 17\right)\cdot 29^{2} + \left(23 a + 18\right)\cdot 29^{3} + \left(12 a + 3\right)\cdot 29^{4} + \left(11 a + 1\right)\cdot 29^{5} + \left(20 a + 21\right)\cdot 29^{6} +O(29^{7})\)
$r_{ 2 }$ |
$=$ |
\( 14 + 20\cdot 29 + 29^{3} + 16\cdot 29^{4} + 9\cdot 29^{5} + 23\cdot 29^{6} +O(29^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 12 a + 1 + \left(7 a + 16\right)\cdot 29 + 27\cdot 29^{2} + \left(22 a + 4\right)\cdot 29^{3} + \left(21 a + 6\right)\cdot 29^{4} + \left(2 a + 1\right)\cdot 29^{5} + \left(26 a + 21\right)\cdot 29^{6} +O(29^{7})\)
| $r_{ 4 }$ |
$=$ |
\( 17 a + 3 + \left(21 a + 12\right)\cdot 29 + \left(28 a + 21\right)\cdot 29^{2} + \left(6 a + 27\right)\cdot 29^{3} + \left(7 a + 5\right)\cdot 29^{4} + \left(26 a + 22\right)\cdot 29^{5} + \left(2 a + 3\right)\cdot 29^{6} +O(29^{7})\)
| $r_{ 5 }$ |
$=$ |
\( 27 + 4\cdot 29 + 28\cdot 29^{2} + 19\cdot 29^{3} + 10\cdot 29^{4} + 7\cdot 29^{5} + 22\cdot 29^{6} +O(29^{7})\)
| $r_{ 6 }$ |
$=$ |
\( 19 a + 18 + \left(13 a + 6\right)\cdot 29 + \left(25 a + 20\right)\cdot 29^{2} + \left(5 a + 14\right)\cdot 29^{3} + \left(16 a + 15\right)\cdot 29^{4} + \left(17 a + 16\right)\cdot 29^{5} + \left(8 a + 24\right)\cdot 29^{6} +O(29^{7})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
$3$ | $2$ | $(1,6)$ | $1$ |
$3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
$6$ | $2$ | $(2,3)(4,5)$ | $1$ |
$6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
$8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
$6$ | $4$ | $(1,4,6,3)$ | $1$ |
$6$ | $4$ | $(1,6)(2,4,5,3)$ | $-1$ |
$8$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.