Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(13725\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 61 \) |
| Artin stem field: | Galois closure of 8.4.211922578125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.305.4t1.a.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-183})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 13x^{6} - 21x^{5} - 135x^{4} + 84x^{3} + 13x^{2} + 672x + 271 \)
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The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 + 5\cdot 89 + 35\cdot 89^{2} + 88\cdot 89^{3} + 6\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 + 85\cdot 89 + 66\cdot 89^{2} + 70\cdot 89^{3} + 24\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 51 + 25\cdot 89^{2} + 77\cdot 89^{3} + 50\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 54 + 42\cdot 89 + 81\cdot 89^{2} + 82\cdot 89^{3} + 87\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 58 + 30\cdot 89 + 56\cdot 89^{2} + 8\cdot 89^{3} + 55\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 64 + 56\cdot 89 + 38\cdot 89^{2} + 87\cdot 89^{3} + 6\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 78 + 18\cdot 89 + 49\cdot 89^{2} + 30\cdot 89^{3} + 13\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 85 + 26\cdot 89 + 3\cdot 89^{2} + 88\cdot 89^{3} + 20\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,5,2,4)(3,6,8,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,5)(3,7,8,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,2,4)(3,7,8,6)$ | $0$ |
| $2$ | $8$ | $(1,8,5,7,2,3,4,6)$ | $0$ |
| $2$ | $8$ | $(1,7,4,8,2,6,5,3)$ | $0$ |
| $2$ | $8$ | $(1,3,4,7,2,8,5,6)$ | $0$ |
| $2$ | $8$ | $(1,7,5,3,2,6,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.