Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(6975\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 31 \) |
| Artin stem field: | Galois closure of 8.0.54731953125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.155.4t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{93})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 17x^{6} - 28x^{5} + 130x^{4} - 182x^{3} + 422x^{2} - 359x + 331 \)
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The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 55\cdot 89 + 59\cdot 89^{2} + 79\cdot 89^{3} + 61\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 + 47\cdot 89 + 30\cdot 89^{2} + 69\cdot 89^{3} + 6\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 + 12\cdot 89 + 33\cdot 89^{2} + 15\cdot 89^{3} + 2\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 52 + 86\cdot 89 + 10\cdot 89^{2} + 3\cdot 89^{3} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 73 + 33\cdot 89 + 22\cdot 89^{2} + 63\cdot 89^{3} + 33\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 80 + 63\cdot 89 + 49\cdot 89^{2} + 9\cdot 89^{3} + 17\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 82 + 74\cdot 89 + 42\cdot 89^{2} + 3\cdot 89^{3} + 55\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 83 + 70\cdot 89 + 17\cdot 89^{2} + 23\cdot 89^{3} + 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,4,2,6)(3,5,7,8)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,2,4)(3,8,7,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,7,5)$ | $0$ |
| $2$ | $8$ | $(1,3,4,5,2,7,6,8)$ | $0$ |
| $2$ | $8$ | $(1,5,6,3,2,8,4,7)$ | $0$ |
| $2$ | $8$ | $(1,7,6,5,2,3,4,8)$ | $0$ |
| $2$ | $8$ | $(1,5,4,7,2,8,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.