Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(6525\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 29 \) |
| Artin stem field: | Galois closure of 8.4.47897578125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.145.4t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-87})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 2x^{6} + 17x^{5} - 80x^{4} + 193x^{3} - 358x^{2} + 226x + 61 \)
|
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 37 + 102\cdot 151 + 25\cdot 151^{2} + 128\cdot 151^{3} + 9\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 64 + 136\cdot 151 + 17\cdot 151^{2} + 72\cdot 151^{3} + 28\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 87 + 144\cdot 151 + 67\cdot 151^{2} + 117\cdot 151^{3} + 76\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 97 + 144\cdot 151 + 96\cdot 151^{2} + 24\cdot 151^{3} + 45\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 102 + 86\cdot 151 + 65\cdot 151^{2} + 67\cdot 151^{3} + 5\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 106 + 86\cdot 151 + 5\cdot 151^{2} + 6\cdot 151^{3} + 69\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 131 + 38\cdot 151 + 149\cdot 151^{2} + 55\cdot 151^{3} + 41\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 132 + 14\cdot 151 + 24\cdot 151^{2} + 132\cdot 151^{3} + 25\cdot 151^{4} +O(151^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $-2$ |
| $2$ | $2$ | $(3,6)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,4,2,8)(3,7,6,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,2,4)(3,5,6,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,2,4)(3,7,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,4,3,2,7,8,6)$ | $0$ |
| $2$ | $8$ | $(1,3,8,5,2,6,4,7)$ | $0$ |
| $2$ | $8$ | $(1,3,4,7,2,6,8,5)$ | $0$ |
| $2$ | $8$ | $(1,7,8,3,2,5,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.