Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(13275\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 59 \) |
| Artin stem field: | Galois closure of 8.8.198253828125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.295.4t1.a.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{177})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 28x^{6} + 47x^{5} + 190x^{4} - 347x^{3} - 358x^{2} + 496x + 331 \)
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The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 + 119\cdot 149 + 126\cdot 149^{2} + 120\cdot 149^{3} + 28\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 46 + 37\cdot 149 + 132\cdot 149^{3} + 90\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 + 73\cdot 149 + 130\cdot 149^{2} + 90\cdot 149^{3} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 82 + 47\cdot 149 + 85\cdot 149^{2} + 113\cdot 149^{3} + 61\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 86 + 32\cdot 149 + 38\cdot 149^{2} + 33\cdot 149^{3} + 148\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 89 + 79\cdot 149 + 85\cdot 149^{2} + 20\cdot 149^{3} + 98\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 94 + 84\cdot 149 + 73\cdot 149^{2} + 70\cdot 149^{3} + 95\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 125 + 121\cdot 149 + 55\cdot 149^{2} + 14\cdot 149^{3} + 72\cdot 149^{4} +O(149^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,6)(3,7)$ | $0$ |
| $1$ | $4$ | $(1,3,6,7)(2,8,4,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,6,3)(2,5,4,8)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,6,7)(2,5,4,8)$ | $0$ |
| $2$ | $8$ | $(1,5,3,2,6,8,7,4)$ | $0$ |
| $2$ | $8$ | $(1,2,7,5,6,4,3,8)$ | $0$ |
| $2$ | $8$ | $(1,8,7,2,6,5,3,4)$ | $0$ |
| $2$ | $8$ | $(1,2,3,8,6,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.