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 8.4.211922578125.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.305.4t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-183})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 13x^{6} - 13x^{5} - 20x^{4} + 358x^{3} + 227x^{2} - 539x + 991 \)
|
The roots of $f$ are computed in $\Q_{ 271 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 117\cdot 271 + 190\cdot 271^{2} + 24\cdot 271^{3} + 53\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 108\cdot 271 + 236\cdot 271^{2} + 18\cdot 271^{3} + 177\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 63 + 208\cdot 271 + 142\cdot 271^{2} + 4\cdot 271^{3} + 75\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 121 + 94\cdot 271 + 129\cdot 271^{2} + 234\cdot 271^{3} + 137\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 154 + 121\cdot 271 + 68\cdot 271^{2} + 136\cdot 271^{3} + 138\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 237 + 61\cdot 271 + 147\cdot 271^{2} + 187\cdot 271^{3} + 34\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 241 + 225\cdot 271 + 183\cdot 271^{2} + 260\cdot 271^{3} + 150\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 246 + 146\cdot 271 + 256\cdot 271^{2} + 216\cdot 271^{3} + 45\cdot 271^{4} +O(271^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $-2$ |
| $2$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,8,7,3)(2,6,5,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,7,8)(2,4,5,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,7,3)(2,4,5,6)$ | $0$ |
| $2$ | $8$ | $(1,2,8,6,7,5,3,4)$ | $0$ |
| $2$ | $8$ | $(1,6,3,2,7,4,8,5)$ | $0$ |
| $2$ | $8$ | $(1,6,8,5,7,4,3,2)$ | $0$ |
| $2$ | $8$ | $(1,5,3,6,7,2,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.