Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:D_4$ |
Conductor: | \(5517\)\(\medspace = 3^{2} \cdot 613 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.4.691039191.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3^2:D_4$ |
Parity: | even |
Determinant: | 1.613.2t1.a.a |
Projective image: | $\SOPlus(4,2)$ |
Projective stem field: | Galois closure of 6.4.691039191.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} - 3x^{4} - 19x^{3} + 27x^{2} + 46x - 21 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 a + 8 + \left(5 a + 1\right)\cdot 19 + \left(a + 15\right)\cdot 19^{2} + \left(16 a + 1\right)\cdot 19^{3} + \left(8 a + 10\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 2 }$ | $=$ | \( 8 a + 3 + \left(13 a + 5\right)\cdot 19 + \left(5 a + 13\right)\cdot 19^{2} + 3 a\cdot 19^{3} + \left(11 a + 3\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 3 }$ | $=$ | \( 11 a + 11 + \left(5 a + 10\right)\cdot 19 + \left(13 a + 5\right)\cdot 19^{2} + \left(15 a + 17\right)\cdot 19^{3} + \left(7 a + 10\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 4 }$ | $=$ | \( 13 + 19 + 12\cdot 19^{2} + 6\cdot 19^{4} +O(19^{5})\) |
$r_{ 5 }$ | $=$ | \( 6 + 3\cdot 19 + 19^{3} + 5\cdot 19^{4} +O(19^{5})\) |
$r_{ 6 }$ | $=$ | \( 9 a + 18 + \left(13 a + 15\right)\cdot 19 + \left(17 a + 10\right)\cdot 19^{2} + \left(2 a + 16\right)\cdot 19^{3} + \left(10 a + 2\right)\cdot 19^{4} +O(19^{5})\) |
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 |
$1$ | $1$ | $()$ | $4$ |
$6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
$6$ | $2$ | $(2,3)$ | $0$ |
$9$ | $2$ | $(1,4)(2,3)$ | $0$ |
$4$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ |
$4$ | $3$ | $(2,3,5)$ | $-2$ |
$18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
$12$ | $6$ | $(1,2,4,3,6,5)$ | $-1$ |
$12$ | $6$ | $(1,4,6)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.