Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.19208000000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.9800.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 5x^{6} + 45x^{4} - 75x^{2} + 50 \)
|
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 + 78\cdot 113 + 107\cdot 113^{2} + 41\cdot 113^{3} + 70\cdot 113^{4} + 109\cdot 113^{5} + 31\cdot 113^{6} +O(113^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 48 + 113 + 65\cdot 113^{2} + 9\cdot 113^{3} + 89\cdot 113^{4} + 72\cdot 113^{5} + 46\cdot 113^{6} +O(113^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 54 + 5\cdot 113 + 98\cdot 113^{2} + 2\cdot 113^{3} + 35\cdot 113^{4} + 111\cdot 113^{5} + 47\cdot 113^{6} +O(113^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 56 + 52\cdot 113 + 16\cdot 113^{2} + 90\cdot 113^{3} + 12\cdot 113^{4} + 52\cdot 113^{5} + 44\cdot 113^{6} +O(113^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 + 60\cdot 113 + 96\cdot 113^{2} + 22\cdot 113^{3} + 100\cdot 113^{4} + 60\cdot 113^{5} + 68\cdot 113^{6} +O(113^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 59 + 107\cdot 113 + 14\cdot 113^{2} + 110\cdot 113^{3} + 77\cdot 113^{4} + 113^{5} + 65\cdot 113^{6} +O(113^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 65 + 111\cdot 113 + 47\cdot 113^{2} + 103\cdot 113^{3} + 23\cdot 113^{4} + 40\cdot 113^{5} + 66\cdot 113^{6} +O(113^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 89 + 34\cdot 113 + 5\cdot 113^{2} + 71\cdot 113^{3} + 42\cdot 113^{4} + 3\cdot 113^{5} + 81\cdot 113^{6} +O(113^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,4)(3,6)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $8$ | $(1,2,6,5,8,7,3,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.