Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(10376\)\(\medspace = 2^{3} \cdot 1297 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.11590971882213376.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Determinant: | 1.10376.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{1297})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 66x^{6} + 1281x^{4} - 4040x^{2} + 9216 \)
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The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 58\cdot 97 + 27\cdot 97^{2} + 34\cdot 97^{3} + 41\cdot 97^{4} + 88\cdot 97^{5} +O(97^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 16 + 36\cdot 97 + 71\cdot 97^{2} + 23\cdot 97^{3} + 22\cdot 97^{4} + 31\cdot 97^{5} +O(97^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 32 + 44\cdot 97 + 86\cdot 97^{2} + 49\cdot 97^{3} + 9\cdot 97^{4} + 80\cdot 97^{5} +O(97^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 44 + 22\cdot 97 + 33\cdot 97^{2} + 39\cdot 97^{3} + 87\cdot 97^{4} + 22\cdot 97^{5} +O(97^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 53 + 74\cdot 97 + 63\cdot 97^{2} + 57\cdot 97^{3} + 9\cdot 97^{4} + 74\cdot 97^{5} +O(97^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 65 + 52\cdot 97 + 10\cdot 97^{2} + 47\cdot 97^{3} + 87\cdot 97^{4} + 16\cdot 97^{5} +O(97^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 81 + 60\cdot 97 + 25\cdot 97^{2} + 73\cdot 97^{3} + 74\cdot 97^{4} + 65\cdot 97^{5} +O(97^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 93 + 38\cdot 97 + 69\cdot 97^{2} + 62\cdot 97^{3} + 55\cdot 97^{4} + 8\cdot 97^{5} +O(97^{6})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ | ✓ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ | |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ | |
| $2$ | $4$ | $(1,7,6,4)(2,3,5,8)$ | $0$ |