Basic invariants
| Dimension: | $4$ |
| Group: | $C_2 \wr C_2\wr C_2$ |
| Conductor: | \(504896\)\(\medspace = 2^{6} \cdot 7^{3} \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.2.4616192.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2 \wr C_2\wr C_2$ |
| Parity: | even |
| Projective image: | $C_2\wr C_2^2$ |
| Projective field: | Galois closure of 8.0.81288256.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$:
\( x^{2} + 82x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 33 a + 84 + \left(53 a + 40\right)\cdot 89 + \left(8 a + 10\right)\cdot 89^{2} + \left(24 a + 35\right)\cdot 89^{3} + \left(14 a + 7\right)\cdot 89^{4} + \left(11 a + 45\right)\cdot 89^{5} + \left(27 a + 79\right)\cdot 89^{6} + \left(11 a + 69\right)\cdot 89^{7} + \left(85 a + 34\right)\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 83 a + 50 + \left(78 a + 74\right)\cdot 89 + \left(7 a + 43\right)\cdot 89^{2} + \left(30 a + 30\right)\cdot 89^{3} + \left(9 a + 64\right)\cdot 89^{4} + \left(79 a + 3\right)\cdot 89^{5} + \left(30 a + 82\right)\cdot 89^{6} + \left(12 a + 46\right)\cdot 89^{7} + \left(88 a + 33\right)\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 55 a + 46 + \left(58 a + 2\right)\cdot 89 + \left(66 a + 31\right)\cdot 89^{2} + \left(41 a + 22\right)\cdot 89^{3} + \left(20 a + 45\right)\cdot 89^{4} + \left(48 a + 10\right)\cdot 89^{5} + 5\cdot 89^{6} + \left(71 a + 33\right)\cdot 89^{7} + \left(65 a + 3\right)\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 + 30\cdot 89 + 50\cdot 89^{2} + 76\cdot 89^{3} + 14\cdot 89^{4} + 73\cdot 89^{5} + 35\cdot 89^{6} + 19\cdot 89^{7} + 38\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 + 81\cdot 89 + 10\cdot 89^{2} + 49\cdot 89^{3} + 72\cdot 89^{4} + 39\cdot 89^{5} + 71\cdot 89^{6} + 55\cdot 89^{7} + 19\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 8 + \left(10 a + 10\right)\cdot 89 + \left(81 a + 20\right)\cdot 89^{2} + \left(58 a + 55\right)\cdot 89^{3} + \left(79 a + 10\right)\cdot 89^{4} + \left(9 a + 14\right)\cdot 89^{5} + \left(58 a + 41\right)\cdot 89^{6} + \left(76 a + 13\right)\cdot 89^{7} + 15\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 a + 75 + \left(30 a + 1\right)\cdot 89 + \left(22 a + 83\right)\cdot 89^{2} + \left(47 a + 69\right)\cdot 89^{3} + \left(68 a + 57\right)\cdot 89^{4} + \left(40 a + 60\right)\cdot 89^{5} + \left(88 a + 49\right)\cdot 89^{6} + \left(17 a + 84\right)\cdot 89^{7} + \left(23 a + 36\right)\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 56 a + 48 + \left(35 a + 25\right)\cdot 89 + \left(80 a + 17\right)\cdot 89^{2} + \left(64 a + 17\right)\cdot 89^{3} + \left(74 a + 83\right)\cdot 89^{4} + \left(77 a + 19\right)\cdot 89^{5} + \left(61 a + 80\right)\cdot 89^{6} + \left(77 a + 32\right)\cdot 89^{7} + \left(3 a + 85\right)\cdot 89^{8} +O(89^{9})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(4,5)$ | $2$ |
| $4$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $2$ |
| $4$ | $2$ | $(1,8)(2,7)(3,6)$ | $-2$ |
| $4$ | $2$ | $(2,3)(6,7)$ | $-2$ |
| $8$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $8$ | $2$ | $(1,8)(2,3)(6,7)$ | $0$ |
| $4$ | $4$ | $(1,8)(2,6,7,3)(4,5)$ | $2$ |
| $4$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(2,3,7,6)$ | $-2$ |
| $8$ | $4$ | $(1,4,8,5)(3,6)$ | $0$ |
| $8$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $8$ | $4$ | $(1,5)(2,6,7,3)(4,8)$ | $0$ |
| $16$ | $4$ | $(1,3)(2,4,7,5)(6,8)$ | $0$ |
| $16$ | $4$ | $(1,2,4,6)(3,8,7,5)$ | $0$ |
| $16$ | $8$ | $(1,6,5,2,8,3,4,7)$ | $0$ |