Basic invariants
Dimension: | $3$ |
Group: | $A_4\times C_2$ |
Conductor: | \(2527\)\(\medspace = 7 \cdot 19^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.4.912247.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4\times C_2$ |
Parity: | odd |
Determinant: | 1.7.2t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.17689.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - 3x^{4} + 5x^{3} - 3x^{2} - x + 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 2 + 4\cdot 11 + 5\cdot 11^{2} + 3\cdot 11^{3} + 5\cdot 11^{4} + 8\cdot 11^{5} + 8\cdot 11^{6} + 9\cdot 11^{7} +O(11^{8})\)
$r_{ 2 }$ |
$=$ |
\( 10 a + 7 + \left(9 a + 7\right)\cdot 11 + \left(7 a + 3\right)\cdot 11^{2} + 2\cdot 11^{3} + 2\cdot 11^{4} + 10\cdot 11^{5} + 7\cdot 11^{6} + \left(7 a + 2\right)\cdot 11^{7} +O(11^{8})\)
| $r_{ 3 }$ |
$=$ |
\( 9 a + 1 + \left(7 a + 6\right)\cdot 11 + \left(9 a + 9\right)\cdot 11^{2} + \left(8 a + 1\right)\cdot 11^{3} + \left(7 a + 5\right)\cdot 11^{4} + \left(9 a + 6\right)\cdot 11^{5} + \left(a + 4\right)\cdot 11^{6} +O(11^{8})\)
| $r_{ 4 }$ |
$=$ |
\( 2 a + 4 + \left(3 a + 6\right)\cdot 11 + \left(a + 7\right)\cdot 11^{2} + \left(2 a + 5\right)\cdot 11^{3} + \left(3 a + 5\right)\cdot 11^{4} + \left(a + 4\right)\cdot 11^{5} + \left(9 a + 2\right)\cdot 11^{6} + \left(10 a + 10\right)\cdot 11^{7} +O(11^{8})\)
| $r_{ 5 }$ |
$=$ |
\( a + 3 + \left(a + 4\right)\cdot 11 + \left(3 a + 3\right)\cdot 11^{2} + \left(10 a + 8\right)\cdot 11^{3} + \left(10 a + 1\right)\cdot 11^{4} + \left(10 a + 10\right)\cdot 11^{5} + \left(10 a + 7\right)\cdot 11^{6} + \left(3 a + 8\right)\cdot 11^{7} +O(11^{8})\)
| $r_{ 6 }$ |
$=$ |
\( 6 + 4\cdot 11 + 3\cdot 11^{2} + 2\cdot 11^{4} + 4\cdot 11^{5} + 11^{6} + 11^{7} +O(11^{8})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
$3$ | $2$ | $(1,6)$ | $1$ |
$3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
$4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
$4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
$4$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
$4$ | $6$ | $(1,3,2,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.