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 8.0.198253828125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.295.4t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{177})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 28x^{6} - 51x^{5} + 255x^{4} - 201x^{3} + 853x^{2} + 222x + 991 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 48\cdot 89 + 85\cdot 89^{2} + 4\cdot 89^{3} + 70\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 31\cdot 89 + 28\cdot 89^{2} + 11\cdot 89^{3} + 52\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 + 70\cdot 89 + 79\cdot 89^{2} + 59\cdot 89^{3} + 23\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 + 59\cdot 89 + 73\cdot 89^{2} + 58\cdot 89^{3} + 68\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 + 27\cdot 89 + 2\cdot 89^{2} + 24\cdot 89^{3} + 39\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 + 72\cdot 89 + 47\cdot 89^{2} + 60\cdot 89^{3} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 79 + 2\cdot 89 + 58\cdot 89^{2} + 31\cdot 89^{3} + 30\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 86 + 43\cdot 89 + 69\cdot 89^{2} + 15\cdot 89^{3} + 71\cdot 89^{4} +O(89^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(2,4)(3,7)$ | $0$ |
| $1$ | $4$ | $(1,6,5,8)(2,3,4,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,5,6)(2,7,4,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,5,8)(2,7,4,3)$ | $0$ |
| $2$ | $8$ | $(1,7,6,2,5,3,8,4)$ | $0$ |
| $2$ | $8$ | $(1,2,8,7,5,4,6,3)$ | $0$ |
| $2$ | $8$ | $(1,2,6,3,5,4,8,7)$ | $0$ |
| $2$ | $8$ | $(1,3,8,2,5,7,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.