Basic invariants
Dimension: | $16$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(138\!\cdots\!984\)\(\medspace = 2^{42} \cdot 11^{12} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.1319329792.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 24T2912 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_3\wr S_3$ |
Projective stem field: | Galois closure of 9.1.1319329792.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} + 3x^{7} - 2x^{6} + 3x^{5} - 4x^{4} - x^{3} + 2x^{2} - 4x + 4 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: \( x^{3} + 6x + 65 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 64 a^{2} + 55 a + 28 + \left(51 a^{2} + 42 a + 32\right)\cdot 67 + \left(66 a^{2} + 63 a + 19\right)\cdot 67^{2} + \left(47 a^{2} + 59 a + 30\right)\cdot 67^{3} + \left(27 a^{2} + 63 a + 13\right)\cdot 67^{4} + \left(2 a^{2} + 2 a + 52\right)\cdot 67^{5} + \left(19 a^{2} + 56 a + 54\right)\cdot 67^{6} + \left(46 a^{2} + 30 a + 55\right)\cdot 67^{7} + \left(43 a^{2} + 34 a + 54\right)\cdot 67^{8} + \left(6 a^{2} + 40 a + 11\right)\cdot 67^{9} +O(67^{10})\)
$r_{ 2 }$ |
$=$ |
\( 64 a + 8 + \left(48 a^{2} + 11 a + 12\right)\cdot 67 + \left(19 a^{2} + 7 a + 3\right)\cdot 67^{2} + \left(63 a^{2} + 27 a + 5\right)\cdot 67^{3} + \left(14 a^{2} + 60 a + 36\right)\cdot 67^{4} + \left(20 a^{2} + 22 a + 9\right)\cdot 67^{5} + \left(19 a^{2} + 31\right)\cdot 67^{6} + \left(59 a^{2} + 33 a + 5\right)\cdot 67^{7} + \left(43 a^{2} + 64 a + 40\right)\cdot 67^{8} + \left(18 a^{2} + 47 a + 55\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 59 a^{2} + 31 a + 43 + \left(38 a^{2} + 47 a + 42\right)\cdot 67 + \left(33 a^{2} + 18 a + 58\right)\cdot 67^{2} + \left(66 a^{2} + 44 a + 17\right)\cdot 67^{3} + \left(52 a^{2} + 47 a + 54\right)\cdot 67^{4} + \left(27 a^{2} + 35 a + 39\right)\cdot 67^{5} + \left(54 a^{2} + 4 a + 37\right)\cdot 67^{6} + \left(10 a^{2} + 59 a + 12\right)\cdot 67^{7} + \left(23 a^{2} + 32 a + 24\right)\cdot 67^{8} + \left(50 a^{2} + 44 a + 48\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 7 a^{2} + 41 a + 1 + \left(59 a^{2} + 19 a + 61\right)\cdot 67 + \left(10 a^{2} + 58 a + 63\right)\cdot 67^{2} + \left(34 a^{2} + 38 a + 41\right)\cdot 67^{3} + \left(61 a^{2} + 60 a + 14\right)\cdot 67^{4} + \left(37 a^{2} + 14 a + 60\right)\cdot 67^{5} + \left(4 a^{2} + 65 a + 63\right)\cdot 67^{6} + \left(11 a^{2} + 27 a + 48\right)\cdot 67^{7} + \left(22 a^{2} + 34 a + 35\right)\cdot 67^{8} + \left(33 a^{2} + 57 a + 51\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 30 a^{2} + 13 a + 5 + \left(48 a^{2} + 13 a + 13\right)\cdot 67 + \left(a^{2} + 5 a + 62\right)\cdot 67^{2} + \left(50 a^{2} + 31 a + 6\right)\cdot 67^{3} + \left(35 a^{2} + 38 a + 63\right)\cdot 67^{4} + \left(45 a^{2} + 57 a + 9\right)\cdot 67^{5} + \left(11 a^{2} + 14 a + 47\right)\cdot 67^{6} + \left(2 a^{2} + 23 a + 34\right)\cdot 67^{7} + \left(3 a^{2} + 25 a + 66\right)\cdot 67^{8} + \left(48 a^{2} + 9 a + 24\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 8 a^{2} + 39 a + 40 + \left(47 a^{2} + 7 a + 8\right)\cdot 67 + \left(13 a^{2} + 41 a + 46\right)\cdot 67^{2} + \left(4 a^{2} + 62 a + 36\right)\cdot 67^{3} + \left(66 a^{2} + 25 a + 39\right)\cdot 67^{4} + \left(18 a^{2} + 8 a + 4\right)\cdot 67^{5} + \left(60 a^{2} + 62 a + 61\right)\cdot 67^{6} + \left(63 a^{2} + 41 a + 23\right)\cdot 67^{7} + \left(66 a^{2} + 36 a + 65\right)\cdot 67^{8} + \left(64 a^{2} + 41 a + 39\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 63 a^{2} + 38 a + 24 + \left(22 a^{2} + 4 a + 50\right)\cdot 67 + \left(56 a^{2} + 12 a + 44\right)\cdot 67^{2} + \left(51 a^{2} + 35 a + 45\right)\cdot 67^{3} + \left(44 a^{2} + 9 a + 14\right)\cdot 67^{4} + \left(26 a^{2} + 49 a + 15\right)\cdot 67^{5} + \left(43 a^{2} + 12 a + 18\right)\cdot 67^{6} + \left(9 a^{2} + 8 a + 43\right)\cdot 67^{7} + \left(a^{2} + 65 a + 18\right)\cdot 67^{8} + \left(27 a^{2} + 35 a + 26\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 35 a^{2} + 6 a + 25 + \left(33 a^{2} + 34 a + 20\right)\cdot 67 + \left(3 a^{2} + 27 a + 2\right)\cdot 67^{2} + \left(17 a^{2} + 28 a + 9\right)\cdot 67^{3} + \left(22 a^{2} + 22 a + 9\right)\cdot 67^{4} + \left(63 a^{2} + 57 a + 14\right)\cdot 67^{5} + \left(42 a^{2} + 32 a + 38\right)\cdot 67^{6} + \left(14 a^{2} + 38 a + 17\right)\cdot 67^{7} + \left(38 a^{2} + 53 a + 6\right)\cdot 67^{8} + \left(16 a^{2} + 11 a + 33\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 2 a^{2} + 48 a + 27 + \left(52 a^{2} + 19 a + 27\right)\cdot 67 + \left(61 a^{2} + 34 a + 34\right)\cdot 67^{2} + \left(66 a^{2} + 7 a + 7\right)\cdot 67^{3} + \left(8 a^{2} + 6 a + 23\right)\cdot 67^{4} + \left(25 a^{2} + 19 a + 62\right)\cdot 67^{5} + \left(12 a^{2} + 19 a + 49\right)\cdot 67^{6} + \left(50 a^{2} + 5 a + 25\right)\cdot 67^{7} + \left(25 a^{2} + 55 a + 23\right)\cdot 67^{8} + \left(2 a^{2} + 45 a + 43\right)\cdot 67^{9} +O(67^{10})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
$1$ | $1$ | $()$ | $16$ |
$9$ | $2$ | $(1,6)$ | $0$ |
$18$ | $2$ | $(1,3)(5,6)(7,9)$ | $0$ |
$27$ | $2$ | $(1,6)(2,4)(3,5)$ | $0$ |
$27$ | $2$ | $(1,6)(3,5)$ | $0$ |
$54$ | $2$ | $(1,6)(2,3)(4,5)(7,8)$ | $0$ |
$6$ | $3$ | $(2,4,8)$ | $-8$ |
$8$ | $3$ | $(1,6,9)(2,4,8)(3,5,7)$ | $-2$ |
$12$ | $3$ | $(1,6,9)(2,4,8)$ | $4$ |
$72$ | $3$ | $(1,3,2)(4,6,5)(7,8,9)$ | $-2$ |
$54$ | $4$ | $(1,5,6,3)(7,9)$ | $0$ |
$162$ | $4$ | $(1,4,6,2)(5,7)(8,9)$ | $0$ |
$36$ | $6$ | $(1,3)(2,4,8)(5,6)(7,9)$ | $0$ |
$36$ | $6$ | $(1,2,6,4,9,8)$ | $0$ |
$36$ | $6$ | $(1,6)(2,4,8)$ | $0$ |
$36$ | $6$ | $(1,6)(2,4,8)(3,5,7)$ | $0$ |
$54$ | $6$ | $(1,6)(2,8,4)(3,5)$ | $0$ |
$72$ | $6$ | $(1,3,6,5,9,7)(2,4,8)$ | $0$ |
$108$ | $6$ | $(1,6)(2,5,4,7,8,3)$ | $0$ |
$216$ | $6$ | $(1,5,4,6,3,2)(7,8,9)$ | $0$ |
$144$ | $9$ | $(1,3,2,6,5,4,9,7,8)$ | $1$ |
$108$ | $12$ | $(1,5,6,3)(2,4,8)(7,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.