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 8.2.8338372875.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 7x^{6} - 21x^{4} + 63x^{3} - 77x^{2} + 60x - 5 \)
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The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 44\cdot 167 + 100\cdot 167^{2} + 50\cdot 167^{3} + 38\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 + 52\cdot 167 + 13\cdot 167^{2} + 51\cdot 167^{3} + 129\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 53 + 151\cdot 167 + 38\cdot 167^{2} + 21\cdot 167^{3} + 65\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 56 + 52\cdot 167 + 123\cdot 167^{2} + 90\cdot 167^{3} + 156\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 101 + 64\cdot 167 + 147\cdot 167^{2} + 122\cdot 167^{3} + 99\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 131 + 141\cdot 167 + 7\cdot 167^{2} + 50\cdot 167^{3} + 94\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 137 + 9\cdot 167 + 32\cdot 167^{2} + 109\cdot 167^{3} + 78\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 150 + 151\cdot 167 + 37\cdot 167^{2} + 5\cdot 167^{3} + 6\cdot 167^{4} +O(167^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $-2$ | |
| $4$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $0$ | |
| $4$ | $2$ | $(1,8)(2,7)(4,6)$ | $0$ | ✓ |
| $2$ | $4$ | $(1,3,8,5)(2,6,4,7)$ | $0$ | |
| $2$ | $8$ | $(1,6,3,4,8,7,5,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | |
| $2$ | $8$ | $(1,4,5,6,8,2,3,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |