Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:C_4$ |
Conductor: | \(2850125\)\(\medspace = 5^{3} \cdot 151^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.324928500625.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3^2:C_4$ |
Parity: | even |
Determinant: | 1.5.2t1.a.a |
Projective image: | $C_3^2:C_4$ |
Projective stem field: | Galois closure of 6.2.324928500625.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} - 99x^{4} + 195x^{3} + 2405x^{2} - 4750x - 26245 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 18 a + 5 + \left(17 a + 7\right)\cdot 19 + \left(3 a + 6\right)\cdot 19^{2} + 9\cdot 19^{3} + \left(11 a + 2\right)\cdot 19^{4} + \left(12 a + 2\right)\cdot 19^{5} + \left(16 a + 14\right)\cdot 19^{6} + \left(16 a + 9\right)\cdot 19^{7} + 6 a\cdot 19^{8} + \left(4 a + 4\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 2 }$ | $=$ | \( 15 a + 5 + \left(17 a + 6\right)\cdot 19 + \left(10 a + 14\right)\cdot 19^{2} + \left(3 a + 9\right)\cdot 19^{3} + \left(10 a + 6\right)\cdot 19^{4} + \left(6 a + 9\right)\cdot 19^{5} + \left(18 a + 17\right)\cdot 19^{6} + \left(11 a + 4\right)\cdot 19^{7} + \left(14 a + 15\right)\cdot 19^{8} + \left(11 a + 14\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 3 }$ | $=$ | \( 4 a + 1 + \left(a + 9\right)\cdot 19 + \left(8 a + 7\right)\cdot 19^{2} + \left(15 a + 2\right)\cdot 19^{3} + \left(8 a + 13\right)\cdot 19^{4} + \left(12 a + 5\right)\cdot 19^{5} + 10\cdot 19^{6} + \left(7 a + 17\right)\cdot 19^{7} + \left(4 a + 17\right)\cdot 19^{8} + \left(7 a + 11\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 4 }$ | $=$ | \( 14 + 3\cdot 19 + 16\cdot 19^{2} + 6\cdot 19^{3} + 18\cdot 19^{4} + 3\cdot 19^{5} + 10\cdot 19^{6} + 15\cdot 19^{7} + 4\cdot 19^{8} + 11\cdot 19^{9} +O(19^{10})\) |
$r_{ 5 }$ | $=$ | \( 11 + 4\cdot 19 + 19^{2} + 4\cdot 19^{3} + 3\cdot 19^{4} + 13\cdot 19^{5} + 5\cdot 19^{6} + 18\cdot 19^{7} + 8\cdot 19^{8} + 13\cdot 19^{9} +O(19^{10})\) |
$r_{ 6 }$ | $=$ | \( a + 4 + \left(a + 7\right)\cdot 19 + \left(15 a + 11\right)\cdot 19^{2} + \left(18 a + 5\right)\cdot 19^{3} + \left(7 a + 13\right)\cdot 19^{4} + \left(6 a + 3\right)\cdot 19^{5} + \left(2 a + 18\right)\cdot 19^{6} + \left(2 a + 9\right)\cdot 19^{7} + \left(12 a + 9\right)\cdot 19^{8} + \left(14 a + 1\right)\cdot 19^{9} +O(19^{10})\) |
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 |
$1$ | $1$ | $()$ | $4$ |
$9$ | $2$ | $(1,5)(2,3)$ | $0$ |
$4$ | $3$ | $(2,3,4)$ | $-2$ |
$4$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
$9$ | $4$ | $(1,3,5,2)(4,6)$ | $0$ |
$9$ | $4$ | $(1,2,5,3)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.