Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(928\!\cdots\!504\)\(\medspace = 2^{36} \cdot 3^{38} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.45349632.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.0.45349632.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} - 14x^{5} + 14x^{4} - 8x^{3} + 7x^{2} - 4x + 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 65 + 52\cdot 79 + 19\cdot 79^{2} + 62\cdot 79^{3} + 61\cdot 79^{4} + 63\cdot 79^{5} + 49\cdot 79^{6} + 5\cdot 79^{7} + 5\cdot 79^{8} + 39\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a + 8 + \left(77 a + 40\right)\cdot 79 + \left(62 a + 36\right)\cdot 79^{2} + \left(75 a + 49\right)\cdot 79^{3} + \left(26 a + 28\right)\cdot 79^{4} + \left(55 a + 57\right)\cdot 79^{5} + \left(76 a + 43\right)\cdot 79^{6} + \left(19 a + 76\right)\cdot 79^{7} + \left(20 a + 58\right)\cdot 79^{8} + \left(3 a + 76\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 61 + 41\cdot 79 + 34\cdot 79^{2} + 16\cdot 79^{3} + 67\cdot 79^{4} + 76\cdot 79^{5} + 50\cdot 79^{6} + 19\cdot 79^{7} + 43\cdot 79^{8} + 63\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 68 a + 62 + \left(55 a + 37\right)\cdot 79 + \left(61 a + 9\right)\cdot 79^{2} + \left(3 a + 29\right)\cdot 79^{3} + \left(2 a + 15\right)\cdot 79^{4} + \left(12 a + 43\right)\cdot 79^{5} + \left(6 a + 31\right)\cdot 79^{6} + \left(18 a + 60\right)\cdot 79^{7} + \left(3 a + 22\right)\cdot 79^{8} + \left(69 a + 34\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 53 a + 34 + \left(a + 12\right)\cdot 79 + \left(16 a + 22\right)\cdot 79^{2} + \left(3 a + 62\right)\cdot 79^{3} + \left(52 a + 58\right)\cdot 79^{4} + \left(23 a + 6\right)\cdot 79^{5} + \left(2 a + 65\right)\cdot 79^{6} + \left(59 a + 19\right)\cdot 79^{7} + \left(58 a + 59\right)\cdot 79^{8} + \left(75 a + 59\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 51 + \left(23 a + 25\right)\cdot 79 + \left(17 a + 15\right)\cdot 79^{2} + \left(75 a + 50\right)\cdot 79^{3} + \left(76 a + 13\right)\cdot 79^{4} + \left(66 a + 53\right)\cdot 79^{5} + \left(72 a + 25\right)\cdot 79^{6} + \left(60 a + 72\right)\cdot 79^{7} + \left(75 a + 7\right)\cdot 79^{8} + \left(9 a + 21\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 27 a + 6 + \left(75 a + 68\right)\cdot 79 + \left(13 a + 40\right)\cdot 79^{2} + \left(13 a + 23\right)\cdot 79^{3} + \left(52 a + 55\right)\cdot 79^{4} + \left(68 a + 38\right)\cdot 79^{5} + \left(22 a + 47\right)\cdot 79^{6} + \left(61 a + 11\right)\cdot 79^{7} + \left(19 a + 1\right)\cdot 79^{8} + \left(57 a + 71\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 52 a + 33 + \left(3 a + 37\right)\cdot 79 + \left(65 a + 58\right)\cdot 79^{2} + \left(65 a + 22\right)\cdot 79^{3} + \left(26 a + 15\right)\cdot 79^{4} + \left(10 a + 55\right)\cdot 79^{5} + \left(56 a + 1\right)\cdot 79^{6} + \left(17 a + 50\right)\cdot 79^{7} + \left(59 a + 38\right)\cdot 79^{8} + \left(21 a + 29\right)\cdot 79^{9} +O(79^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,7)(5,8)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,5)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,7)(5,8)$ | $0$ |
| $16$ | $3$ | $(1,4,6)$ | $0$ |
| $64$ | $3$ | $(1,4,6)(5,7,8)$ | $0$ |
| $12$ | $4$ | $(2,5,7,8)$ | $0$ |
| $36$ | $4$ | $(1,3,4,6)(2,5,7,8)$ | $-2$ |
| $36$ | $4$ | $(1,3,4,6)(2,7)(5,8)$ | $0$ |
| $72$ | $4$ | $(1,2,4,7)(3,5,6,8)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,5,7,8)$ | $2$ |
| $144$ | $4$ | $(1,5,3,2)(4,7)(6,8)$ | $0$ |
| $48$ | $6$ | $(1,6,4)(2,7)(5,8)$ | $0$ |
| $96$ | $6$ | $(1,3)(5,8,7)$ | $0$ |
| $192$ | $6$ | $(1,5,4,7,6,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,5,4,7,6,8)$ | $0$ |
| $96$ | $12$ | $(1,4,6)(2,5,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.