Basic invariants
Dimension: | $9$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(337032380416\)\(\medspace = 2^{17} \cdot 137^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 8.2.45091246208.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 16T1294 |
Parity: | odd |
Projective image: | $S_4\wr C_2$ |
Projective field: | Galois closure of 8.2.45091246208.1 |
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 $\mathbb{Q}_{ 19 }$:
\( x^{3} + 4x + 17 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 a^{2} + 3 a + \left(3 a^{2} + 18 a + 16\right)\cdot 19 + \left(18 a^{2} + 2 a + 11\right)\cdot 19^{2} + \left(9 a^{2} + 13 a + 6\right)\cdot 19^{3} + \left(18 a^{2} + 12 a + 13\right)\cdot 19^{4} + \left(10 a^{2} + 17 a + 6\right)\cdot 19^{5} + \left(8 a^{2} + 18 a\right)\cdot 19^{6} + \left(a^{2} + 9 a + 8\right)\cdot 19^{7} + \left(12 a^{2} + 4 a + 14\right)\cdot 19^{8} + \left(2 a^{2} + 12 a + 8\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 2 }$ | $=$ | \( 4 + 18\cdot 19 + 14\cdot 19^{2} + 2\cdot 19^{3} + 13\cdot 19^{4} + 10\cdot 19^{5} + 10\cdot 19^{6} + 6\cdot 19^{7} + 15\cdot 19^{8} + 13\cdot 19^{9} +O(19^{10})\) |
$r_{ 3 }$ | $=$ | \( 10 a^{2} + 12 a + 8\cdot 19 + \left(14 a^{2} + 16 a + 13\right)\cdot 19^{2} + \left(a + 13\right)\cdot 19^{3} + \left(11 a^{2} + a + 18\right)\cdot 19^{4} + \left(14 a^{2} + 9 a + 9\right)\cdot 19^{5} + \left(18 a^{2} + 8\right)\cdot 19^{6} + \left(17 a^{2} + 15 a + 1\right)\cdot 19^{7} + \left(11 a^{2} + 15 a + 14\right)\cdot 19^{8} + \left(3 a^{2} + 7 a + 17\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 4 }$ | $=$ | \( 8 a^{2} + 8 a + 18 + \left(2 a^{2} + 11 a + 13\right)\cdot 19 + \left(14 a^{2} + 2 a + 8\right)\cdot 19^{2} + \left(12 a^{2} + 9 a + 1\right)\cdot 19^{3} + \left(12 a^{2} + 6 a + 9\right)\cdot 19^{4} + \left(2 a^{2} + 17 a + 11\right)\cdot 19^{5} + \left(5 a^{2} + 16 a + 8\right)\cdot 19^{6} + \left(15 a^{2} + 17 a + 8\right)\cdot 19^{7} + \left(13 a + 5\right)\cdot 19^{8} + \left(10 a^{2} + 9 a + 9\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 5 }$ | $=$ | \( 2 a^{2} + 8 a + 2 + \left(a^{2} + 11 a + 4\right)\cdot 19 + \left(12 a^{2} + 18 a + 3\right)\cdot 19^{2} + \left(7 a^{2} + 5 a + 13\right)\cdot 19^{3} + \left(11 a^{2} + 2 a + 5\right)\cdot 19^{4} + \left(6 a^{2} + 8 a + 9\right)\cdot 19^{5} + \left(18 a^{2} + 17 a + 18\right)\cdot 19^{6} + \left(9 a^{2} + 14 a + 6\right)\cdot 19^{7} + \left(5 a^{2} + 5\right)\cdot 19^{8} + \left(6 a^{2} + 17 a + 18\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 6 }$ | $=$ | \( 9 a^{2} + 3 a + 8 + \left(15 a^{2} + 15 a + 4\right)\cdot 19 + \left(11 a^{2} + 16 a + 15\right)\cdot 19^{2} + \left(17 a^{2} + 3 a + 1\right)\cdot 19^{3} + \left(13 a^{2} + 10 a + 6\right)\cdot 19^{4} + \left(9 a^{2} + 12 a + 11\right)\cdot 19^{5} + \left(14 a^{2} + 3 a + 14\right)\cdot 19^{6} + \left(12 a^{2} + 5 a + 1\right)\cdot 19^{7} + \left(12 a^{2} + 4 a + 18\right)\cdot 19^{8} + \left(2 a^{2} + 11 a + 14\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 7 }$ | $=$ | \( 10 + 15\cdot 19 + 10\cdot 19^{2} + 2\cdot 19^{3} + 17\cdot 19^{4} + 5\cdot 19^{5} + 15\cdot 19^{6} + 19^{7} + 9\cdot 19^{8} + 14\cdot 19^{9} +O(19^{10})\) |
$r_{ 8 }$ | $=$ | \( 18 a^{2} + 4 a + 15 + \left(14 a^{2} + 14\right)\cdot 19 + \left(5 a^{2} + 16\right)\cdot 19^{2} + \left(8 a^{2} + 4 a + 14\right)\cdot 19^{3} + \left(8 a^{2} + 5 a + 11\right)\cdot 19^{4} + \left(12 a^{2} + 11 a + 10\right)\cdot 19^{5} + \left(10 a^{2} + 18 a + 18\right)\cdot 19^{6} + \left(18 a^{2} + 12 a + 2\right)\cdot 19^{7} + \left(13 a^{2} + 17 a + 13\right)\cdot 19^{8} + \left(12 a^{2} + 17 a + 16\right)\cdot 19^{9} +O(19^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $9$ |
$6$ | $2$ | $(1,3)(2,8)$ | $-3$ |
$9$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $1$ |
$12$ | $2$ | $(1,2)$ | $3$ |
$24$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $3$ |
$36$ | $2$ | $(1,2)(4,5)$ | $1$ |
$36$ | $2$ | $(1,2)(4,6)(5,7)$ | $-1$ |
$16$ | $3$ | $(1,3,8)$ | $0$ |
$64$ | $3$ | $(1,3,8)(4,6,7)$ | $0$ |
$12$ | $4$ | $(1,2,3,8)$ | $-3$ |
$36$ | $4$ | $(1,2,3,8)(4,5,6,7)$ | $1$ |
$36$ | $4$ | $(1,3)(2,8)(4,5,6,7)$ | $1$ |
$72$ | $4$ | $(1,6,3,4)(2,7,8,5)$ | $-1$ |
$72$ | $4$ | $(1,2)(4,5,6,7)$ | $-1$ |
$144$ | $4$ | $(1,5,2,4)(3,6)(7,8)$ | $1$ |
$48$ | $6$ | $(1,8,3)(4,6)(5,7)$ | $0$ |
$96$ | $6$ | $(1,3,8)(4,5)$ | $0$ |
$192$ | $6$ | $(1,6,3,7,8,4)(2,5)$ | $0$ |
$144$ | $8$ | $(1,5,2,6,3,7,8,4)$ | $-1$ |
$96$ | $12$ | $(1,3,8)(4,5,6,7)$ | $0$ |