Basic invariants
Dimension: | $9$ |
Group: | $(A_4\wr C_2):C_2$ |
Conductor: | \(723183538176\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 11^{3} \cdot 17^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.4.1584788925456.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | 12T165 |
Parity: | even |
Determinant: | 1.561.2t1.a.a |
Projective image: | $\PGOPlus(4,3)$ |
Projective stem field: | Galois closure of 8.4.1584788925456.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} + 4x^{6} - 25x^{4} + 50x^{3} + 16 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 49 a^{2} + 59 a + 10 + \left(11 a^{2} + 21 a + 12\right)\cdot 67 + \left(59 a^{2} + 21\right)\cdot 67^{2} + \left(34 a^{2} + 58 a + 45\right)\cdot 67^{3} + \left(66 a^{2} + 35 a + 49\right)\cdot 67^{4} + \left(45 a^{2} + 62 a + 44\right)\cdot 67^{5} + \left(41 a^{2} + 2 a + 64\right)\cdot 67^{6} + \left(60 a^{2} + 38 a + 29\right)\cdot 67^{7} + \left(11 a^{2} + 29 a + 39\right)\cdot 67^{8} + \left(22 a^{2} + 5 a + 36\right)\cdot 67^{9} +O(67^{10})\)
$r_{ 2 }$ |
$=$ |
\( 46 + 28\cdot 67 + 50\cdot 67^{2} + 52\cdot 67^{3} + 38\cdot 67^{4} + 25\cdot 67^{5} + 20\cdot 67^{6} + 16\cdot 67^{7} + 43\cdot 67^{8} + 35\cdot 67^{9} +O(67^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 22 a^{2} + 7 a + 51 + \left(22 a^{2} + 53 a + 12\right)\cdot 67 + \left(32 a^{2} + 25 a + 23\right)\cdot 67^{2} + \left(5 a^{2} + 40 a + 4\right)\cdot 67^{3} + \left(49 a^{2} + 39 a + 27\right)\cdot 67^{4} + \left(60 a^{2} + 10 a + 33\right)\cdot 67^{5} + \left(23 a^{2} + 16 a + 66\right)\cdot 67^{6} + \left(47 a^{2} + 47 a + 49\right)\cdot 67^{7} + \left(19 a^{2} + 54 a + 19\right)\cdot 67^{8} + \left(36 a^{2} + 10 a + 66\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 34 a^{2} + 9 a + 17 + \left(15 a^{2} + 35 a + 27\right)\cdot 67 + \left(62 a^{2} + 35 a + 33\right)\cdot 67^{2} + \left(55 a^{2} + 49 a + 62\right)\cdot 67^{3} + \left(10 a^{2} + 28 a + 27\right)\cdot 67^{4} + \left(2 a^{2} + 43 a + 3\right)\cdot 67^{5} + \left(53 a^{2} + 17 a + 43\right)\cdot 67^{6} + \left(9 a^{2} + 13 a + 27\right)\cdot 67^{7} + \left(28 a^{2} + 11 a + 37\right)\cdot 67^{8} + \left(12 a^{2} + 5 a + 64\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 24 + 37\cdot 67 + 43\cdot 67^{2} + 14\cdot 67^{3} + 46\cdot 67^{4} + 15\cdot 67^{5} + 38\cdot 67^{6} + 34\cdot 67^{7} + 24\cdot 67^{8} + 22\cdot 67^{9} +O(67^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 22 a^{2} + 15 a + 51 + \left(48 a^{2} + 8 a + 49\right)\cdot 67 + \left(34 a^{2} + 52 a + 32\right)\cdot 67^{2} + \left(65 a^{2} + 58 a + 43\right)\cdot 67^{3} + \left(14 a^{2} + 57 a + 24\right)\cdot 67^{4} + \left(9 a^{2} + 7 a + 28\right)\cdot 67^{5} + \left(3 a^{2} + 50\right)\cdot 67^{6} + \left(28 a^{2} + 17 a + 39\right)\cdot 67^{7} + \left(30 a^{2} + 38 a + 62\right)\cdot 67^{8} + \left(21 a^{2} + 8 a + 6\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 51 a^{2} + 66 a + 18 + \left(39 a^{2} + 9 a + 57\right)\cdot 67 + \left(12 a^{2} + 31 a + 35\right)\cdot 67^{2} + \left(43 a^{2} + 26 a + 11\right)\cdot 67^{3} + \left(56 a^{2} + 2 a + 10\right)\cdot 67^{4} + \left(18 a^{2} + 28 a + 3\right)\cdot 67^{5} + \left(39 a^{2} + 46 a + 55\right)\cdot 67^{6} + \left(63 a^{2} + 15 a + 41\right)\cdot 67^{7} + \left(26 a^{2} + 26 a + 32\right)\cdot 67^{8} + \left(32 a^{2} + 56 a + 10\right)\cdot 67^{9} +O(67^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 23 a^{2} + 45 a + 55 + \left(63 a^{2} + 5 a + 42\right)\cdot 67 + \left(66 a^{2} + 56 a + 27\right)\cdot 67^{2} + \left(62 a^{2} + 34 a + 33\right)\cdot 67^{3} + \left(2 a^{2} + 36 a + 43\right)\cdot 67^{4} + \left(64 a^{2} + 48 a + 46\right)\cdot 67^{5} + \left(39 a^{2} + 50 a + 63\right)\cdot 67^{6} + \left(58 a^{2} + 2 a + 27\right)\cdot 67^{7} + \left(16 a^{2} + 41 a + 8\right)\cdot 67^{8} + \left(9 a^{2} + 47 a + 25\right)\cdot 67^{9} +O(67^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $9$ |
$6$ | $2$ | $(1,5)(4,7)$ | $-3$ |
$9$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $1$ |
$12$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $3$ |
$12$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $3$ |
$36$ | $2$ | $(3,6)(5,7)$ | $1$ |
$16$ | $3$ | $(2,8,3)$ | $0$ |
$32$ | $3$ | $(1,7,5)(2,8,6)$ | $0$ |
$32$ | $3$ | $(1,7,5)(2,8,3)$ | $0$ |
$36$ | $4$ | $(1,6,5,2)(3,4,8,7)$ | $-1$ |
$36$ | $4$ | $(1,7,5,4)(2,3,6,8)$ | $1$ |
$36$ | $4$ | $(1,3,5,8)(2,4,6,7)$ | $-1$ |
$72$ | $4$ | $(1,7,5,4)(6,8)$ | $-1$ |
$48$ | $6$ | $(1,5)(2,8,3)(4,7)$ | $0$ |
$96$ | $6$ | $(1,8,7,6,5,2)(3,4)$ | $0$ |
$96$ | $6$ | $(1,3,7,2,5,8)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.