Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(4275\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 19 \) |
| Artin stem field: | Galois closure of 8.0.20560078125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.95.4t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{57})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 13x^{6} + x^{5} + 55x^{4} + 46x^{3} + 133x^{2} + 73x + 211 \)
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The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 24\cdot 59 + 5\cdot 59^{2} + 32\cdot 59^{3} + 29\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 48\cdot 59 + 45\cdot 59^{2} + 14\cdot 59^{3} + 14\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 39\cdot 59 + 15\cdot 59^{2} + 58\cdot 59^{3} + 29\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 + 6\cdot 59 + 51\cdot 59^{2} + 12\cdot 59^{3} + 44\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 + 42\cdot 59 + 3\cdot 59^{2} + 45\cdot 59^{3} + 15\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 39 + 2\cdot 59 + 8\cdot 59^{2} + 35\cdot 59^{3} + 12\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 44 + 9\cdot 59 + 18\cdot 59^{2} + 42\cdot 59^{3} + 28\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 58 + 3\cdot 59 + 29\cdot 59^{2} + 54\cdot 59^{3} + 59^{4} +O(59^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $1$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)(5,6,8,7)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,4,2)(5,6,8,7)$ | $0$ |
| $2$ | $8$ | $(1,8,3,6,4,5,2,7)$ | $0$ |
| $2$ | $8$ | $(1,6,2,8,4,7,3,5)$ | $0$ |
| $2$ | $8$ | $(1,5,2,6,4,8,3,7)$ | $0$ |
| $2$ | $8$ | $(1,6,3,5,4,7,2,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.