Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(975\)\(\medspace = 3 \cdot 5^{2} \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.2780578125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.39.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.2925.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 6x^{6} + x^{5} + 9x^{4} - 27x^{3} + 79x^{2} - 6x + 21 \)
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The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 68 + 290\cdot 337 + 236\cdot 337^{2} + 95\cdot 337^{3} + 111\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 98 + 155\cdot 337 + 18\cdot 337^{2} + 45\cdot 337^{3} + 284\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 137 + 51\cdot 337 + 122\cdot 337^{2} + 167\cdot 337^{3} + 32\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 140 + 111\cdot 337 + 112\cdot 337^{2} + 6\cdot 337^{3} + 233\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 162 + 203\cdot 337 + 141\cdot 337^{2} + 16\cdot 337^{3} + 248\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 172 + 89\cdot 337 + 68\cdot 337^{2} + 125\cdot 337^{3} + 336\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 263 + 255\cdot 337 + 134\cdot 337^{2} + 143\cdot 337^{3} + 196\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 311 + 190\cdot 337 + 176\cdot 337^{2} + 74\cdot 337^{3} + 243\cdot 337^{4} +O(337^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(2,6)(3,4)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,8,5,7)(2,3,4,6)$ | $0$ |
| $2$ | $8$ | $(1,4,8,6,5,2,7,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,6,7,4,5,3,8,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.