Basic invariants
Dimension: | $3$ |
Group: | $A_4\times C_2$ |
Conductor: | \(1377\)\(\medspace = 3^{4} \cdot 17 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.111537.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4\times C_2$ |
Parity: | even |
Determinant: | 1.17.2t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.23409.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 3x^{5} + 3x^{4} + x^{3} - 3x^{2} + 3x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( a + 8 + \left(17 a + 3\right)\cdot 19 + \left(5 a + 8\right)\cdot 19^{2} + \left(12 a + 11\right)\cdot 19^{3} + \left(7 a + 12\right)\cdot 19^{4} + \left(11 a + 7\right)\cdot 19^{5} + \left(7 a + 10\right)\cdot 19^{6} +O(19^{7})\) |
$r_{ 2 }$ | $=$ | \( 9 a + 2 + \left(11 a + 1\right)\cdot 19 + 13\cdot 19^{2} + \left(6 a + 7\right)\cdot 19^{3} + \left(17 a + 2\right)\cdot 19^{4} + \left(11 a + 3\right)\cdot 19^{5} + 16 a\cdot 19^{6} +O(19^{7})\) |
$r_{ 3 }$ | $=$ | \( 18 a + 9 + a\cdot 19 + \left(13 a + 16\right)\cdot 19^{2} + \left(6 a + 17\right)\cdot 19^{3} + \left(11 a + 7\right)\cdot 19^{4} + \left(7 a + 11\right)\cdot 19^{5} + \left(11 a + 6\right)\cdot 19^{6} +O(19^{7})\) |
$r_{ 4 }$ | $=$ | \( 6 + 13\cdot 19 + 10\cdot 19^{2} + 17\cdot 19^{3} + 18\cdot 19^{4} + 7\cdot 19^{5} + 13\cdot 19^{6} +O(19^{7})\) |
$r_{ 5 }$ | $=$ | \( 5 + 16\cdot 19 + 6\cdot 19^{2} + 8\cdot 19^{3} + 19^{4} + 10\cdot 19^{5} + 2\cdot 19^{6} +O(19^{7})\) |
$r_{ 6 }$ | $=$ | \( 10 a + 11 + \left(7 a + 3\right)\cdot 19 + \left(18 a + 2\right)\cdot 19^{2} + \left(12 a + 13\right)\cdot 19^{3} + \left(a + 13\right)\cdot 19^{4} + \left(7 a + 16\right)\cdot 19^{5} + \left(2 a + 4\right)\cdot 19^{6} +O(19^{7})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-3$ |
$3$ | $2$ | $(1,3)$ | $1$ |
$3$ | $2$ | $(1,3)(2,6)$ | $-1$ |
$4$ | $3$ | $(1,2,4)(3,6,5)$ | $0$ |
$4$ | $3$ | $(1,4,2)(3,5,6)$ | $0$ |
$4$ | $6$ | $(1,6,5,3,2,4)$ | $0$ |
$4$ | $6$ | $(1,4,2,3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.