Basic invariants
| Dimension: | $4$ |
| Group: | $((C_8 : C_2):C_2):C_2$ |
| Conductor: | \(128787625\)\(\medspace = 5^{3} \cdot 101^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1578125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $((C_8 : C_2):C_2):C_2$ |
| Parity: | even |
| Determinant: | 1.505.2t1.a.a |
| Projective image: | $C_2^3:C_4$ |
| Projective stem field: | Galois closure of 8.4.65037750625.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 3x^{5} - x^{4} - 3x^{3} + 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 701 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 144 + 248\cdot 701 + 197\cdot 701^{2} + 198\cdot 701^{3} + 233\cdot 701^{4} + 228\cdot 701^{5} +O(701^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 165 + 624\cdot 701 + 174\cdot 701^{2} + 575\cdot 701^{3} + 106\cdot 701^{4} + 612\cdot 701^{5} +O(701^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 200 + 53\cdot 701 + 412\cdot 701^{2} + 676\cdot 701^{3} + 358\cdot 701^{4} + 525\cdot 701^{5} +O(701^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 258 + 378\cdot 701 + 392\cdot 701^{2} + 481\cdot 701^{3} + 397\cdot 701^{4} + 610\cdot 701^{5} +O(701^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 354 + 477\cdot 701 + 167\cdot 701^{2} + 127\cdot 701^{3} + 142\cdot 701^{4} + 672\cdot 701^{5} +O(701^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 424 + 501\cdot 701 + 511\cdot 701^{2} + 304\cdot 701^{3} + 430\cdot 701^{4} + 462\cdot 701^{5} +O(701^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 577 + 273\cdot 701 + 300\cdot 701^{2} + 417\cdot 701^{3} + 340\cdot 701^{4} + 100\cdot 701^{5} +O(701^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 684 + 246\cdot 701 + 647\cdot 701^{2} + 22\cdot 701^{3} + 93\cdot 701^{4} + 293\cdot 701^{5} +O(701^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $-4$ |
| $2$ | $2$ | $(2,8)(3,5)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,8)(6,7)$ | $2$ |
| $4$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $0$ |
| $4$ | $2$ | $(2,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,4)$ | $-2$ |
| $4$ | $4$ | $(1,7,4,6)(2,3,8,5)$ | $0$ |
| $8$ | $4$ | $(1,6)(2,3,8,5)(4,7)$ | $0$ |
| $8$ | $4$ | $(1,2,6,3)(4,8,7,5)$ | $0$ |
| $8$ | $4$ | $(1,3,6,2)(4,5,7,8)$ | $0$ |
| $8$ | $8$ | $(1,8,7,5,4,2,6,3)$ | $0$ |
| $8$ | $8$ | $(1,5,6,8,4,3,7,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.