Basic invariants
| Dimension: | $4$ |
| Group: | $C_2 \wr S_4$ |
| Conductor: | \(385308179\)\(\medspace = 17 \cdot 283^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1361513.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2 \wr S_4$ |
| Parity: | odd |
| Determinant: | 1.4811.2t1.a.a |
| Projective image: | $C_2^3:S_4$ |
| Projective stem field: | Galois closure of 8.4.6689113369.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 2x^{6} - 3x^{5} + 3x^{4} - 3x^{3} + 2x^{2} - x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{2} + 78x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 54 a + 44 + \left(39 a + 5\right)\cdot 79 + \left(47 a + 68\right)\cdot 79^{2} + \left(27 a + 42\right)\cdot 79^{3} + \left(47 a + 59\right)\cdot 79^{4} + \left(61 a + 6\right)\cdot 79^{5} + \left(31 a + 18\right)\cdot 79^{6} + \left(72 a + 49\right)\cdot 79^{7} + \left(57 a + 76\right)\cdot 79^{8} + \left(36 a + 67\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 60 a + 72 + \left(15 a + 11\right)\cdot 79 + \left(53 a + 40\right)\cdot 79^{2} + \left(25 a + 39\right)\cdot 79^{3} + \left(57 a + 25\right)\cdot 79^{4} + \left(73 a + 11\right)\cdot 79^{5} + \left(62 a + 51\right)\cdot 79^{6} + \left(24 a + 1\right)\cdot 79^{7} + \left(45 a + 35\right)\cdot 79^{8} + \left(76 a + 71\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 72 + 6\cdot 79 + 46\cdot 79^{2} + 26\cdot 79^{3} + 55\cdot 79^{4} + 62\cdot 79^{5} + 35\cdot 79^{6} + 36\cdot 79^{7} + 19\cdot 79^{8} + 68\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 a + 19 + \left(39 a + 70\right)\cdot 79 + \left(31 a + 75\right)\cdot 79^{2} + \left(51 a + 22\right)\cdot 79^{3} + 31 a\cdot 79^{4} + \left(17 a + 21\right)\cdot 79^{5} + \left(47 a + 67\right)\cdot 79^{6} + \left(6 a + 10\right)\cdot 79^{7} + \left(21 a + 62\right)\cdot 79^{8} + \left(42 a + 46\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 78 a + 46 + \left(28 a + 38\right)\cdot 79 + \left(40 a + 42\right)\cdot 79^{2} + \left(74 a + 27\right)\cdot 79^{3} + \left(32 a + 48\right)\cdot 79^{4} + \left(48 a + 33\right)\cdot 79^{5} + \left(52 a + 56\right)\cdot 79^{6} + \left(53 a + 19\right)\cdot 79^{7} + \left(21 a + 57\right)\cdot 79^{8} + \left(56 a + 69\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 45 + 67\cdot 79 + 69\cdot 79^{2} + 3\cdot 79^{3} + 63\cdot 79^{4} + 24\cdot 79^{5} + 65\cdot 79^{6} + 55\cdot 79^{7} + 63\cdot 79^{8} + 21\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 19 a + 53 + \left(63 a + 46\right)\cdot 79 + \left(25 a + 77\right)\cdot 79^{2} + \left(53 a + 11\right)\cdot 79^{3} + \left(21 a + 57\right)\cdot 79^{4} + \left(5 a + 27\right)\cdot 79^{5} + \left(16 a + 40\right)\cdot 79^{6} + \left(54 a + 42\right)\cdot 79^{7} + \left(33 a + 55\right)\cdot 79^{8} + \left(2 a + 23\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( a + 45 + \left(50 a + 68\right)\cdot 79 + \left(38 a + 53\right)\cdot 79^{2} + \left(4 a + 61\right)\cdot 79^{3} + \left(46 a + 6\right)\cdot 79^{4} + \left(30 a + 49\right)\cdot 79^{5} + \left(26 a + 60\right)\cdot 79^{6} + \left(25 a + 20\right)\cdot 79^{7} + \left(57 a + 25\right)\cdot 79^{8} + \left(22 a + 25\right)\cdot 79^{9} +O(79^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $4$ | $2$ | $(1,8)$ | $2$ |
| $4$ | $2$ | $(1,8)(2,7)(4,5)$ | $-2$ |
| $6$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $12$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $12$ | $2$ | $(3,4)(5,6)$ | $-2$ |
| $12$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $2$ |
| $24$ | $2$ | $(1,8)(3,4)(5,6)$ | $0$ |
| $32$ | $3$ | $(1,4,2)(5,7,8)$ | $1$ |
| $12$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $12$ | $4$ | $(3,5,6,4)$ | $-2$ |
| $12$ | $4$ | $(1,7,8,2)(3,6)(4,5)$ | $2$ |
| $24$ | $4$ | $(1,6,8,3)(2,4)(5,7)$ | $0$ |
| $24$ | $4$ | $(1,8)(3,5,6,4)$ | $0$ |
| $48$ | $4$ | $(1,4,3,2)(5,6,7,8)$ | $0$ |
| $32$ | $6$ | $(1,5,7,8,4,2)$ | $1$ |
| $32$ | $6$ | $(1,4,2)(3,6)(5,7,8)$ | $-1$ |
| $32$ | $6$ | $(1,5,7,8,4,2)(3,6)$ | $-1$ |
| $48$ | $8$ | $(1,5,6,7,8,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.