Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(579\!\cdots\!179\)\(\medspace = 3^{27} \cdot 97^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.21512615283.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.291.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.21512615283.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 7x^{6} - 2x^{5} + 8x^{4} + 7x^{3} + 28x^{2} - 20x + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{3} + 9x + 76 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 56\cdot 79 + 43\cdot 79^{2} + 38\cdot 79^{3} + 40\cdot 79^{4} + 64\cdot 79^{5} + 2\cdot 79^{6} + 78\cdot 79^{7} + 65\cdot 79^{8} + 35\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 68 a^{2} + 70 a + 38 + \left(58 a^{2} + 34 a + 18\right)\cdot 79 + \left(4 a^{2} + 29 a + 40\right)\cdot 79^{2} + \left(43 a^{2} + 38 a + 8\right)\cdot 79^{3} + \left(35 a + 16\right)\cdot 79^{4} + \left(46 a^{2} + 36 a + 70\right)\cdot 79^{5} + \left(50 a^{2} + 31 a + 12\right)\cdot 79^{6} + \left(31 a + 4\right)\cdot 79^{7} + \left(8 a^{2} + 59 a + 26\right)\cdot 79^{8} + \left(29 a^{2} + 72 a + 57\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a^{2} + 33 a + 42 + \left(48 a^{2} + 36 a + 68\right)\cdot 79 + \left(32 a^{2} + 41 a + 54\right)\cdot 79^{2} + \left(52 a^{2} + 42 a + 76\right)\cdot 79^{3} + \left(30 a^{2} + 57 a + 36\right)\cdot 79^{4} + \left(15 a^{2} + 40 a + 50\right)\cdot 79^{5} + \left(52 a^{2} + 69 a + 76\right)\cdot 79^{6} + \left(49 a^{2} + 20 a + 12\right)\cdot 79^{7} + \left(45 a^{2} + 70 a + 45\right)\cdot 79^{8} + \left(66 a^{2} + 10 a + 61\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 69 a^{2} + 19 a + 44 + \left(65 a^{2} + 32 a + 60\right)\cdot 79 + \left(23 a^{2} + 25 a + 75\right)\cdot 79^{2} + \left(76 a^{2} + 54 a + 49\right)\cdot 79^{3} + \left(66 a^{2} + 60 a + 19\right)\cdot 79^{4} + \left(14 a^{2} + 32 a + 41\right)\cdot 79^{5} + \left(26 a^{2} + 46 a + 24\right)\cdot 79^{6} + \left(76 a^{2} + 68 a + 63\right)\cdot 79^{7} + \left(27 a^{2} + 51 a + 66\right)\cdot 79^{8} + \left(14 a^{2} + 47\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 a^{2} + 2 a + 33 + \left(3 a^{2} + 68 a + 38\right)\cdot 79 + \left(39 a^{2} + 43 a + 14\right)\cdot 79^{2} + \left(44 a^{2} + 6 a + 29\right)\cdot 79^{3} + \left(2 a^{2} + 2 a + 26\right)\cdot 79^{4} + \left(21 a^{2} + 7 a + 5\right)\cdot 79^{5} + \left(37 a^{2} + 23 a + 66\right)\cdot 79^{6} + \left(20 a^{2} + 60 a + 74\right)\cdot 79^{7} + \left(56 a^{2} + 5 a + 29\right)\cdot 79^{8} + \left(36 a^{2} + 47 a + 40\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 31 a^{2} + 44 a + 40 + \left(27 a^{2} + 53 a + 23\right)\cdot 79 + \left(7 a^{2} + 72 a + 61\right)\cdot 79^{2} + \left(61 a^{2} + 29 a + 49\right)\cdot 79^{3} + \left(45 a^{2} + 19 a + 48\right)\cdot 79^{4} + \left(42 a^{2} + 31 a + 55\right)\cdot 79^{5} + \left(68 a^{2} + 65 a + 16\right)\cdot 79^{6} + \left(8 a^{2} + 76 a + 5\right)\cdot 79^{7} + \left(56 a^{2} + 2 a + 29\right)\cdot 79^{8} + \left(54 a^{2} + 21 a + 69\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 43 + 27\cdot 79 + 27\cdot 79^{2} + 2\cdot 79^{3} + 46\cdot 79^{4} + 46\cdot 79^{5} + 77\cdot 79^{6} + 64\cdot 79^{7} + 53\cdot 79^{8} + 65\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 21 a^{2} + 69 a + 72 + \left(33 a^{2} + 11 a + 22\right)\cdot 79 + \left(50 a^{2} + 24 a + 77\right)\cdot 79^{2} + \left(38 a^{2} + 65 a + 60\right)\cdot 79^{3} + \left(11 a^{2} + 61 a + 2\right)\cdot 79^{4} + \left(18 a^{2} + 9 a + 61\right)\cdot 79^{5} + \left(2 a^{2} + a + 38\right)\cdot 79^{6} + \left(2 a^{2} + 58 a + 12\right)\cdot 79^{7} + \left(43 a^{2} + 46 a + 78\right)\cdot 79^{8} + \left(35 a^{2} + 5 a + 16\right)\cdot 79^{9} +O(79^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(3,6)(5,7)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $2$ |
| $12$ | $2$ | $(3,5)$ | $0$ |
| $24$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,2)(3,5)$ | $-2$ |
| $36$ | $2$ | $(1,4)(2,8)(3,5)$ | $0$ |
| $16$ | $3$ | $(5,7,6)$ | $0$ |
| $64$ | $3$ | $(2,8,4)(5,7,6)$ | $0$ |
| $12$ | $4$ | $(3,5,6,7)$ | $0$ |
| $36$ | $4$ | $(1,2,4,8)(3,5,6,7)$ | $-2$ |
| $36$ | $4$ | $(1,2,4,8)(3,6)(5,7)$ | $0$ |
| $72$ | $4$ | $(1,3,4,6)(2,5,8,7)$ | $0$ |
| $72$ | $4$ | $(1,2,4,8)(3,5)$ | $2$ |
| $144$ | $4$ | $(1,3,2,5)(4,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,4)(2,8)(5,6,7)$ | $0$ |
| $96$ | $6$ | $(2,8,4)(3,5)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,5,8,7,4,6)$ | $0$ |
| $144$ | $8$ | $(1,3,2,5,4,6,8,7)$ | $0$ |
| $96$ | $12$ | $(1,2,4,8)(5,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.