Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(2475\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 11 \) |
| Artin stem field: | Galois closure of 8.8.6891328125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.55.4t1.a.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{33})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 13x^{6} + 17x^{5} + 40x^{4} - 62x^{3} - 13x^{2} + 31x + 1 \)
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The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 17\cdot 31 + 9\cdot 31^{2} + 11\cdot 31^{3} + 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 26\cdot 31 + 10\cdot 31^{2} + 16\cdot 31^{3} + 25\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 7\cdot 31 + 13\cdot 31^{3} + 16\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 30\cdot 31 + 20\cdot 31^{2} + 24\cdot 31^{3} + 24\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 + 23\cdot 31 + 10\cdot 31^{3} + 25\cdot 31^{4} +O(31^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 28 + 14\cdot 31 + 2\cdot 31^{2} + 17\cdot 31^{3} + 4\cdot 31^{4} +O(31^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 29 + 23\cdot 31 + 16\cdot 31^{2} + 8\cdot 31^{3} + 31^{4} +O(31^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 30 + 10\cdot 31 + 23\cdot 31^{3} + 24\cdot 31^{4} +O(31^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,6)(4,7)$ | $0$ |
| $1$ | $4$ | $(1,7,6,4)(2,3,8,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,6,7)(2,5,8,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,6,4)(2,5,8,3)$ | $0$ |
| $2$ | $8$ | $(1,2,7,3,6,8,4,5)$ | $0$ |
| $2$ | $8$ | $(1,3,4,2,6,5,7,8)$ | $0$ |
| $2$ | $8$ | $(1,8,4,3,6,2,7,5)$ | $0$ |
| $2$ | $8$ | $(1,3,7,8,6,5,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.