Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(117\)\(\medspace = 3^{2} \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.1601613.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.13.4t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.6591.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{6} - 3x^{5} + 3x^{4} + 3x^{3} - 2x^{2} + 1 \)
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The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 48 + 52\cdot 139 + 37\cdot 139^{2} + 86\cdot 139^{3} + 30\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 55 + 24\cdot 139 + 70\cdot 139^{2} + 32\cdot 139^{3} + 93\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 74 + 93\cdot 139 + 61\cdot 139^{2} + 116\cdot 139^{3} + 51\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 77 + 83\cdot 139 + 15\cdot 139^{2} + 95\cdot 139^{3} + 98\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 90 + 116\cdot 139 + 20\cdot 139^{2} + 25\cdot 139^{3} + 118\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 93 + 94\cdot 139 + 130\cdot 139^{2} + 28\cdot 139^{3} + 62\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 122 + 72\cdot 139 + 124\cdot 139^{2} + 31\cdot 139^{3} + 123\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 136 + 17\cdot 139 + 95\cdot 139^{2} + 117\cdot 139^{4} +O(139^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-2$ | |
| $2$ | $2$ | $(3,4)(5,7)$ | $0$ | |
| $4$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $0$ | ✓ |
| $1$ | $4$ | $(1,6,2,8)(3,5,4,7)$ | $-2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,8,2,6)(3,7,4,5)$ | $2 \zeta_{4}$ | |
| $2$ | $4$ | $(3,7,4,5)$ | $\zeta_{4} + 1$ | |
| $2$ | $4$ | $(3,5,4,7)$ | $-\zeta_{4} + 1$ | |
| $2$ | $4$ | $(1,2)(3,5,4,7)(6,8)$ | $-\zeta_{4} - 1$ | |
| $2$ | $4$ | $(1,2)(3,7,4,5)(6,8)$ | $\zeta_{4} - 1$ | |
| $2$ | $4$ | $(1,6,2,8)(3,7,4,5)$ | $0$ | |
| $4$ | $4$ | $(1,7,2,5)(3,8,4,6)$ | $0$ | |
| $4$ | $8$ | $(1,3,8,7,2,4,6,5)$ | $0$ | |
| $4$ | $8$ | $(1,7,6,3,2,5,8,4)$ | $0$ |