Basic invariants
Dimension: | $9$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(410011373853\)\(\medspace = 3^{4} \cdot 17^{3} \cdot 101^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 8.4.683285301.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 16T1294 |
Parity: | even |
Projective image: | $S_4\wr C_2$ |
Projective field: | Galois closure of 8.4.683285301.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a + 16 + \left(10 a + 53\right)\cdot 67 + \left(52 a + 18\right)\cdot 67^{2} + \left(37 a + 59\right)\cdot 67^{3} + \left(11 a + 46\right)\cdot 67^{4} + \left(53 a + 64\right)\cdot 67^{5} + \left(41 a + 4\right)\cdot 67^{6} + \left(4 a + 14\right)\cdot 67^{7} + \left(64 a + 30\right)\cdot 67^{8} + \left(3 a + 45\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 2 }$ | $=$ | \( 32 + 48\cdot 67 + 12\cdot 67^{2} + 63\cdot 67^{3} + 5\cdot 67^{4} + 8\cdot 67^{5} + 22\cdot 67^{6} + 45\cdot 67^{7} + 64\cdot 67^{8} + 40\cdot 67^{9} +O(67^{10})\) |
$r_{ 3 }$ | $=$ | \( 54 a + 3 + \left(18 a + 58\right)\cdot 67 + \left(42 a + 8\right)\cdot 67^{2} + \left(43 a + 60\right)\cdot 67^{3} + \left(27 a + 36\right)\cdot 67^{4} + \left(23 a + 63\right)\cdot 67^{5} + \left(32 a + 50\right)\cdot 67^{6} + \left(53 a + 22\right)\cdot 67^{7} + \left(40 a + 16\right)\cdot 67^{8} + \left(11 a + 27\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 4 }$ | $=$ | \( 19 + 54\cdot 67 + 30\cdot 67^{2} + 6\cdot 67^{3} + 40\cdot 67^{4} + 36\cdot 67^{5} + 60\cdot 67^{6} + 55\cdot 67^{7} + 47\cdot 67^{8} + 42\cdot 67^{9} +O(67^{10})\) |
$r_{ 5 }$ | $=$ | \( 12 a + 10 + \left(34 a + 16\right)\cdot 67 + \left(12 a + 36\right)\cdot 67^{2} + \left(32 a + 40\right)\cdot 67^{3} + \left(33 a + 5\right)\cdot 67^{4} + \left(31 a + 38\right)\cdot 67^{5} + \left(31 a + 13\right)\cdot 67^{6} + \left(2 a + 45\right)\cdot 67^{7} + \left(63 a + 60\right)\cdot 67^{8} + \left(48 a + 7\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 6 }$ | $=$ | \( 59 a + 48 + \left(56 a + 18\right)\cdot 67 + \left(14 a + 16\right)\cdot 67^{2} + \left(29 a + 24\right)\cdot 67^{3} + \left(55 a + 55\right)\cdot 67^{4} + \left(13 a + 64\right)\cdot 67^{5} + \left(25 a + 51\right)\cdot 67^{6} + \left(62 a + 57\right)\cdot 67^{7} + \left(2 a + 13\right)\cdot 67^{8} + \left(63 a + 64\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 7 }$ | $=$ | \( 13 a + 18 + \left(48 a + 12\right)\cdot 67 + \left(24 a + 25\right)\cdot 67^{2} + \left(23 a + 58\right)\cdot 67^{3} + \left(39 a + 36\right)\cdot 67^{4} + \left(43 a + 62\right)\cdot 67^{5} + \left(34 a + 22\right)\cdot 67^{6} + \left(13 a + 3\right)\cdot 67^{7} + \left(26 a + 59\right)\cdot 67^{8} + \left(55 a + 32\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 8 }$ | $=$ | \( 55 a + 58 + \left(32 a + 6\right)\cdot 67 + \left(54 a + 52\right)\cdot 67^{2} + \left(34 a + 22\right)\cdot 67^{3} + \left(33 a + 40\right)\cdot 67^{4} + \left(35 a + 63\right)\cdot 67^{5} + \left(35 a + 40\right)\cdot 67^{6} + \left(64 a + 23\right)\cdot 67^{7} + \left(3 a + 42\right)\cdot 67^{8} + \left(18 a + 6\right)\cdot 67^{9} +O(67^{10})\) |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $9$ |
$6$ | $2$ | $(1,6)(3,7)$ | $-3$ |
$9$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $1$ |
$12$ | $2$ | $(1,3)$ | $3$ |
$24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $3$ |
$36$ | $2$ | $(1,3)(2,4)$ | $1$ |
$36$ | $2$ | $(1,3)(2,5)(4,8)$ | $-1$ |
$16$ | $3$ | $(3,6,7)$ | $0$ |
$64$ | $3$ | $(3,6,7)(4,5,8)$ | $0$ |
$12$ | $4$ | $(1,3,6,7)$ | $-3$ |
$36$ | $4$ | $(1,3,6,7)(2,4,5,8)$ | $1$ |
$36$ | $4$ | $(1,6)(2,4,5,8)(3,7)$ | $1$ |
$72$ | $4$ | $(1,5,6,2)(3,8,7,4)$ | $-1$ |
$72$ | $4$ | $(1,3)(2,4,5,8)$ | $-1$ |
$144$ | $4$ | $(1,4,3,2)(5,6)(7,8)$ | $1$ |
$48$ | $6$ | $(2,5)(3,7,6)(4,8)$ | $0$ |
$96$ | $6$ | $(1,3)(4,8,5)$ | $0$ |
$192$ | $6$ | $(1,2)(3,5,6,8,7,4)$ | $0$ |
$144$ | $8$ | $(1,4,3,5,6,8,7,2)$ | $-1$ |
$96$ | $12$ | $(2,4,5,8)(3,6,7)$ | $0$ |