Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(3536\)\(\medspace = 2^{4} \cdot 13 \cdot 17 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.45968.3 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.884.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{13}, \sqrt{-17})\) |
Defining polynomial
$f(x)$ | $=$ |
\( x^{4} + 7x^{2} - 17 \)
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The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 7 + 41\cdot 101 + 62\cdot 101^{2} + 60\cdot 101^{3} + 48\cdot 101^{4} +O(101^{5})\)
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$r_{ 2 }$ | $=$ |
\( 34 + 30\cdot 101 + 90\cdot 101^{2} + 92\cdot 101^{3} + 49\cdot 101^{4} +O(101^{5})\)
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$r_{ 3 }$ | $=$ |
\( 67 + 70\cdot 101 + 10\cdot 101^{2} + 8\cdot 101^{3} + 51\cdot 101^{4} +O(101^{5})\)
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$r_{ 4 }$ | $=$ |
\( 94 + 59\cdot 101 + 38\cdot 101^{2} + 40\cdot 101^{3} + 52\cdot 101^{4} +O(101^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.