Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:C_4$ |
Conductor: | \(990125\)\(\medspace = 5^{3} \cdot 89^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.2.39213900625.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3^2:C_4$ |
Parity: | even |
Projective image: | $C_3^2:C_4$ |
Projective field: | Galois closure of 6.2.39213900625.1 |
Galois action
Roots of defining polynomial
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:
$r_{ 1 }$ | $=$ | \( 7 a + 6 + \left(13 a + 2\right)\cdot 19 + 2 a\cdot 19^{2} + \left(16 a + 18\right)\cdot 19^{3} + \left(2 a + 10\right)\cdot 19^{4} + \left(15 a + 10\right)\cdot 19^{5} + \left(18 a + 6\right)\cdot 19^{6} + \left(9 a + 18\right)\cdot 19^{7} + 14 a\cdot 19^{8} + \left(2 a + 15\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 2 }$ | $=$ | \( 16 + 5\cdot 19 + 12\cdot 19^{3} + 2\cdot 19^{4} + 10\cdot 19^{5} + 15\cdot 19^{6} + 16\cdot 19^{7} + 3\cdot 19^{8} + 15\cdot 19^{9} +O(19^{10})\) |
$r_{ 3 }$ | $=$ | \( 14 a + \left(3 a + 10\right)\cdot 19 + 17\cdot 19^{2} + \left(15 a + 13\right)\cdot 19^{3} + \left(2 a + 8\right)\cdot 19^{4} + \left(a + 1\right)\cdot 19^{5} + 11 a\cdot 19^{6} + \left(12 a + 3\right)\cdot 19^{7} + \left(18 a + 7\right)\cdot 19^{8} + \left(16 a + 17\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 4 }$ | $=$ | \( 5 a + 14 + \left(15 a + 18\right)\cdot 19 + \left(18 a + 13\right)\cdot 19^{2} + \left(3 a + 9\right)\cdot 19^{3} + \left(16 a + 15\right)\cdot 19^{4} + \left(17 a + 18\right)\cdot 19^{5} + \left(7 a + 9\right)\cdot 19^{6} + \left(6 a + 4\right)\cdot 19^{7} + 13\cdot 19^{8} + \left(2 a + 15\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 5 }$ | $=$ | \( 11 + 11\cdot 19 + 16\cdot 19^{2} + 9\cdot 19^{3} + 2\cdot 19^{4} + 12\cdot 19^{5} + 14\cdot 19^{6} + 4\cdot 19^{7} + 7\cdot 19^{8} + 9\cdot 19^{9} +O(19^{10})\) |
$r_{ 6 }$ | $=$ | \( 12 a + 13 + \left(5 a + 8\right)\cdot 19 + \left(16 a + 8\right)\cdot 19^{2} + \left(2 a + 12\right)\cdot 19^{3} + \left(16 a + 16\right)\cdot 19^{4} + \left(3 a + 3\right)\cdot 19^{5} + 10\cdot 19^{6} + \left(9 a + 9\right)\cdot 19^{7} + \left(4 a + 5\right)\cdot 19^{8} + \left(16 a + 3\right)\cdot 19^{9} +O(19^{10})\) |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $4$ |
$9$ | $2$ | $(1,2)(3,4)$ | $0$ |
$4$ | $3$ | $(1,2,6)$ | $-2$ |
$4$ | $3$ | $(1,2,6)(3,4,5)$ | $1$ |
$9$ | $4$ | $(1,4,2,3)(5,6)$ | $0$ |
$9$ | $4$ | $(1,3,2,4)(5,6)$ | $0$ |