Basic invariants
| Dimension: | $4$ |
| Group: | $C_2 \wr C_2\wr C_2$ |
| Conductor: | \(269568\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1513893888.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2 \wr C_2\wr C_2$ |
| Parity: | even |
| Determinant: | 1.13.2t1.a.a |
| Projective image: | $C_2\wr C_2^2$ |
| Projective stem field: | Galois closure of 8.0.504631296.7 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 15x^{4} - 24x^{2} - 12 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 67 + \left(70 a + 2\right)\cdot 71 + \left(56 a + 49\right)\cdot 71^{2} + \left(56 a + 42\right)\cdot 71^{3} + \left(7 a + 20\right)\cdot 71^{4} + \left(14 a + 25\right)\cdot 71^{5} + \left(23 a + 19\right)\cdot 71^{6} + \left(58 a + 24\right)\cdot 71^{7} + \left(63 a + 36\right)\cdot 71^{8} + \left(36 a + 30\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 a + 43 + \left(28 a + 56\right)\cdot 71 + \left(2 a + 11\right)\cdot 71^{2} + \left(6 a + 66\right)\cdot 71^{3} + \left(61 a + 12\right)\cdot 71^{4} + \left(35 a + 30\right)\cdot 71^{5} + \left(60 a + 28\right)\cdot 71^{6} + \left(56 a + 44\right)\cdot 71^{7} + \left(55 a + 43\right)\cdot 71^{8} + \left(48 a + 14\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 a + 25 + \left(67 a + 26\right)\cdot 71 + \left(35 a + 33\right)\cdot 71^{2} + \left(41 a + 47\right)\cdot 71^{3} + \left(58 a + 68\right)\cdot 71^{4} + \left(4 a + 59\right)\cdot 71^{5} + \left(43 a + 65\right)\cdot 71^{6} + \left(18 a + 2\right)\cdot 71^{7} + \left(40 a + 40\right)\cdot 71^{8} + \left(4 a + 15\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 32 + 50\cdot 71 + 68\cdot 71^{2} + 63\cdot 71^{3} + 17\cdot 71^{4} + 68\cdot 71^{5} + 47\cdot 71^{6} + 31\cdot 71^{7} + 51\cdot 71^{8} + 36\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 67 a + 4 + 68\cdot 71 + \left(14 a + 21\right)\cdot 71^{2} + \left(14 a + 28\right)\cdot 71^{3} + \left(63 a + 50\right)\cdot 71^{4} + \left(56 a + 45\right)\cdot 71^{5} + \left(47 a + 51\right)\cdot 71^{6} + \left(12 a + 46\right)\cdot 71^{7} + \left(7 a + 34\right)\cdot 71^{8} + \left(34 a + 40\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 43 a + 28 + \left(42 a + 14\right)\cdot 71 + \left(68 a + 59\right)\cdot 71^{2} + \left(64 a + 4\right)\cdot 71^{3} + \left(9 a + 58\right)\cdot 71^{4} + \left(35 a + 40\right)\cdot 71^{5} + \left(10 a + 42\right)\cdot 71^{6} + \left(14 a + 26\right)\cdot 71^{7} + \left(15 a + 27\right)\cdot 71^{8} + \left(22 a + 56\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 25 a + 46 + \left(3 a + 44\right)\cdot 71 + \left(35 a + 37\right)\cdot 71^{2} + \left(29 a + 23\right)\cdot 71^{3} + \left(12 a + 2\right)\cdot 71^{4} + \left(66 a + 11\right)\cdot 71^{5} + \left(27 a + 5\right)\cdot 71^{6} + \left(52 a + 68\right)\cdot 71^{7} + \left(30 a + 30\right)\cdot 71^{8} + \left(66 a + 55\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 39 + 20\cdot 71 + 2\cdot 71^{2} + 7\cdot 71^{3} + 53\cdot 71^{4} + 2\cdot 71^{5} + 23\cdot 71^{6} + 39\cdot 71^{7} + 19\cdot 71^{8} + 34\cdot 71^{9} +O(71^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-4$ |
| $2$ | $2$ | $(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(3,7)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)$ | $2$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)$ | $2$ |
| $4$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $-2$ |
| $8$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $8$ | $2$ | $(1,6)(2,5)(3,7)$ | $0$ |
| $4$ | $4$ | $(1,5)(2,6)(3,4,7,8)$ | $2$ |
| $4$ | $4$ | $(1,6,5,2)(3,8,7,4)$ | $0$ |
| $4$ | $4$ | $(1,2,5,6)$ | $-2$ |
| $8$ | $4$ | $(2,6)(3,4,7,8)$ | $0$ |
| $8$ | $4$ | $(1,6)(2,5)(3,4,7,8)$ | $0$ |
| $8$ | $4$ | $(1,8,5,4)(2,3,6,7)$ | $0$ |
| $16$ | $4$ | $(1,7,2,4)(3,6,8,5)$ | $0$ |
| $16$ | $4$ | $(1,8,5,4)(2,3)(6,7)$ | $0$ |
| $16$ | $8$ | $(1,3,6,8,5,7,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.