Basic invariants
Dimension: | $9$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(19087679828125\)\(\medspace = 5^{6} \cdot 1069^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.4.6529513515605.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 18T272 |
Parity: | even |
Determinant: | 1.1069.2t1.a.a |
Projective image: | $S_4\wr C_2$ |
Projective stem field: | Galois closure of 8.4.6529513515605.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + 11x^{6} - 39x^{5} + 87x^{4} - 178x^{3} + 108x^{2} + 56x + 5 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 18 a + 25 + \left(24 a + 32\right)\cdot 37 + \left(30 a + 29\right)\cdot 37^{2} + \left(22 a + 26\right)\cdot 37^{3} + \left(3 a + 29\right)\cdot 37^{4} + \left(7 a + 32\right)\cdot 37^{5} + \left(5 a + 22\right)\cdot 37^{6} + \left(9 a + 26\right)\cdot 37^{7} + \left(18 a + 32\right)\cdot 37^{8} + \left(3 a + 31\right)\cdot 37^{9} +O(37^{10})\)
$r_{ 2 }$ |
$=$ |
\( 20 a + 29 + \left(15 a + 35\right)\cdot 37 + \left(14 a + 10\right)\cdot 37^{2} + \left(20 a + 20\right)\cdot 37^{3} + \left(7 a + 6\right)\cdot 37^{4} + \left(7 a + 18\right)\cdot 37^{5} + \left(24 a + 36\right)\cdot 37^{6} + \left(3 a + 17\right)\cdot 37^{7} + \left(29 a + 8\right)\cdot 37^{8} + \left(30 a + 34\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 18 + 11\cdot 37 + 11\cdot 37^{2} + 19\cdot 37^{3} + 20\cdot 37^{4} + 36\cdot 37^{6} + 19\cdot 37^{7} + 19\cdot 37^{8} + 5\cdot 37^{9} +O(37^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 17 a + 35 + \left(21 a + 3\right)\cdot 37 + \left(22 a + 16\right)\cdot 37^{2} + \left(16 a + 13\right)\cdot 37^{3} + \left(29 a + 16\right)\cdot 37^{4} + \left(29 a + 2\right)\cdot 37^{5} + \left(12 a + 15\right)\cdot 37^{6} + \left(33 a + 8\right)\cdot 37^{7} + \left(7 a + 10\right)\cdot 37^{8} + \left(6 a + 17\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 21 + 30\cdot 37 + 4\cdot 37^{2} + 8\cdot 37^{3} + 12\cdot 37^{4} + 9\cdot 37^{5} + 20\cdot 37^{6} + 23\cdot 37^{7} + 11\cdot 37^{8} + 8\cdot 37^{9} +O(37^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 12 a + 31 + \left(2 a + 35\right)\cdot 37 + \left(29 a + 8\right)\cdot 37^{2} + \left(30 a + 13\right)\cdot 37^{3} + \left(14 a + 6\right)\cdot 37^{4} + \left(36 a + 22\right)\cdot 37^{5} + \left(4 a + 35\right)\cdot 37^{6} + \left(35 a + 2\right)\cdot 37^{7} + \left(32 a + 10\right)\cdot 37^{8} + \left(23 a + 17\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 25 a + 5 + \left(34 a + 33\right)\cdot 37 + \left(7 a + 11\right)\cdot 37^{2} + \left(6 a + 33\right)\cdot 37^{3} + \left(22 a + 34\right)\cdot 37^{4} + 4\cdot 37^{5} + \left(32 a + 19\right)\cdot 37^{6} + \left(a + 27\right)\cdot 37^{7} + \left(4 a + 32\right)\cdot 37^{8} + \left(13 a + 5\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 19 a + 23 + \left(12 a + 1\right)\cdot 37 + \left(6 a + 17\right)\cdot 37^{2} + \left(14 a + 13\right)\cdot 37^{3} + \left(33 a + 21\right)\cdot 37^{4} + \left(29 a + 20\right)\cdot 37^{5} + \left(31 a + 36\right)\cdot 37^{6} + \left(27 a + 20\right)\cdot 37^{7} + \left(18 a + 22\right)\cdot 37^{8} + \left(33 a + 27\right)\cdot 37^{9} +O(37^{10})\)
| |
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 value |
$1$ | $1$ | $()$ | $9$ |
$6$ | $2$ | $(1,4)(2,8)$ | $-3$ |
$9$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $1$ |
$12$ | $2$ | $(3,5)$ | $-3$ |
$24$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $3$ |
$36$ | $2$ | $(1,2)(3,5)$ | $1$ |
$36$ | $2$ | $(1,4)(2,8)(3,5)$ | $1$ |
$16$ | $3$ | $(3,6,7)$ | $0$ |
$64$ | $3$ | $(2,4,8)(3,6,7)$ | $0$ |
$12$ | $4$ | $(1,2,4,8)$ | $3$ |
$36$ | $4$ | $(1,2,4,8)(3,5,6,7)$ | $1$ |
$36$ | $4$ | $(1,4)(2,8)(3,5,6,7)$ | $-1$ |
$72$ | $4$ | $(1,6,4,3)(2,7,8,5)$ | $-1$ |
$72$ | $4$ | $(1,2,4,8)(3,5)$ | $-1$ |
$144$ | $4$ | $(1,3,2,5)(4,6)(7,8)$ | $-1$ |
$48$ | $6$ | $(1,4)(2,8)(3,7,6)$ | $0$ |
$96$ | $6$ | $(2,8,4)(3,5)$ | $0$ |
$192$ | $6$ | $(1,5)(2,6,4,7,8,3)$ | $0$ |
$144$ | $8$ | $(1,5,2,6,4,7,8,3)$ | $1$ |
$96$ | $12$ | $(1,2,4,8)(3,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.