Basic invariants
| Dimension: | $4$ |
| Group: | $(C_4^2 : C_2):C_2$ |
| Conductor: | \(4460544\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.28262006784.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $(C_4^2 : C_2):C_2$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^2\wr C_2$ |
| Projective stem field: | Galois closure of 8.0.4081676544.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 2x^{4} - 11 \)
|
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 150\cdot 181 + 112\cdot 181^{2} + 105\cdot 181^{3} + 36\cdot 181^{4} + 107\cdot 181^{5} + 21\cdot 181^{6} + 25\cdot 181^{7} + 156\cdot 181^{8} + 145\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 + 74\cdot 181 + 26\cdot 181^{2} + 62\cdot 181^{3} + 46\cdot 181^{4} + 71\cdot 181^{5} + 4\cdot 181^{6} + 167\cdot 181^{7} + 155\cdot 181^{8} + 27\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 61 + 97\cdot 181 + 136\cdot 181^{2} + 39\cdot 181^{3} + 82\cdot 181^{4} + 30\cdot 181^{5} + 16\cdot 181^{6} + 69\cdot 181^{7} + 158\cdot 181^{8} + 81\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 73 + 112\cdot 181 + 118\cdot 181^{2} + 61\cdot 181^{3} + 133\cdot 181^{4} + 121\cdot 181^{5} + 104\cdot 181^{6} + 39\cdot 181^{7} + 61\cdot 181^{8} + 169\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 108 + 68\cdot 181 + 62\cdot 181^{2} + 119\cdot 181^{3} + 47\cdot 181^{4} + 59\cdot 181^{5} + 76\cdot 181^{6} + 141\cdot 181^{7} + 119\cdot 181^{8} + 11\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 120 + 83\cdot 181 + 44\cdot 181^{2} + 141\cdot 181^{3} + 98\cdot 181^{4} + 150\cdot 181^{5} + 164\cdot 181^{6} + 111\cdot 181^{7} + 22\cdot 181^{8} + 99\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 152 + 106\cdot 181 + 154\cdot 181^{2} + 118\cdot 181^{3} + 134\cdot 181^{4} + 109\cdot 181^{5} + 176\cdot 181^{6} + 13\cdot 181^{7} + 25\cdot 181^{8} + 153\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 173 + 30\cdot 181 + 68\cdot 181^{2} + 75\cdot 181^{3} + 144\cdot 181^{4} + 73\cdot 181^{5} + 159\cdot 181^{6} + 155\cdot 181^{7} + 24\cdot 181^{8} + 35\cdot 181^{9} +O(181^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $8$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $4$ | $4$ | $(1,7,8,2)(3,6)(4,5)$ | $-2$ |
| $4$ | $4$ | $(1,7,8,2)$ | $2$ |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $8$ | $8$ | $(1,6,7,5,8,3,2,4)$ | $0$ |
| $8$ | $8$ | $(1,6,7,4,8,3,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.