Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(735\)\(\medspace = 3 \cdot 5 \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.8338372875.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.15.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.15435.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 7x^{6} - 21x^{5} + 49x^{4} - 84x^{3} + 84x^{2} - 45x + 9 \)
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The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 30 + 228\cdot 257 + 86\cdot 257^{2} + 24\cdot 257^{3} + 119\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 + 94\cdot 257 + 59\cdot 257^{2} + 21\cdot 257^{3} + 239\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 + 14\cdot 257 + 211\cdot 257^{2} + 236\cdot 257^{3} + 231\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 57 + 217\cdot 257^{2} + 159\cdot 257^{3} + 186\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 119 + 248\cdot 257 + 42\cdot 257^{2} + 84\cdot 257^{3} + 26\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 123 + 24\cdot 257 + 121\cdot 257^{2} + 65\cdot 257^{3} + 61\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 126 + 229\cdot 257 + 115\cdot 257^{2} + 11\cdot 257^{3} + 11\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 239 + 188\cdot 257 + 173\cdot 257^{2} + 167\cdot 257^{3} + 152\cdot 257^{4} +O(257^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,3,7,8)(2,4,5,6)$ | $0$ |
| $2$ | $8$ | $(1,5,3,6,7,2,8,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,6,8,5,7,4,3,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.