Normalized defining polynomial
\( x^{16} - 8x^{14} + 32x^{12} - 72x^{10} + 102x^{8} - 88x^{6} + 32x^{4} + 8x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(288230376151711744\) \(\medspace = 2^{58}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(12.34\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $2^{29/8}\approx 12.337686603263526$ | ||
Ramified primes: | \(2\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{1148}a^{14}-\frac{69}{1148}a^{12}+\frac{223}{1148}a^{10}+\frac{101}{1148}a^{8}+\frac{255}{1148}a^{6}+\frac{429}{1148}a^{4}-\frac{307}{1148}a^{2}+\frac{367}{1148}$, $\frac{1}{1148}a^{15}-\frac{69}{1148}a^{13}-\frac{16}{287}a^{11}-\frac{1}{4}a^{10}-\frac{93}{574}a^{9}-\frac{1}{4}a^{8}+\frac{255}{1148}a^{7}+\frac{429}{1148}a^{5}-\frac{5}{287}a^{3}+\frac{1}{4}a^{2}-\frac{247}{574}a+\frac{1}{4}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{27}{287} a^{14} + \frac{851}{1148} a^{12} - \frac{855}{287} a^{10} + \frac{7747}{1148} a^{8} - \frac{2867}{287} a^{6} + \frac{10781}{1148} a^{4} - \frac{1469}{287} a^{2} + \frac{257}{1148} \) (order $8$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{177}{574}a^{15}-\frac{2901}{1148}a^{13}+\frac{2946}{287}a^{11}-\frac{27099}{1148}a^{9}+\frac{9796}{287}a^{7}-\frac{35545}{1148}a^{5}+\frac{7653}{574}a^{3}+\frac{481}{1148}a$, $\frac{79}{1148}a^{14}-\frac{143}{287}a^{12}+\frac{2119}{1148}a^{10}-\frac{2181}{574}a^{8}+\frac{5795}{1148}a^{6}-\frac{2427}{574}a^{4}+\frac{1003}{1148}a^{2}+\frac{145}{287}$, $\frac{33}{1148}a^{14}-\frac{67}{287}a^{12}+\frac{1045}{1148}a^{10}-\frac{530}{287}a^{8}+\frac{2101}{1148}a^{6}+\frac{47}{574}a^{4}-\frac{2095}{1148}a^{2}+\frac{459}{574}$, $\frac{49}{164}a^{15}+\frac{5}{82}a^{14}-\frac{97}{41}a^{13}-\frac{75}{164}a^{12}+\frac{769}{82}a^{11}+\frac{303}{164}a^{10}-\frac{3415}{164}a^{9}-\frac{178}{41}a^{8}+\frac{4787}{164}a^{7}+\frac{289}{41}a^{6}-\frac{1028}{41}a^{5}-\frac{1163}{164}a^{4}+\frac{781}{82}a^{3}+\frac{579}{164}a^{2}+\frac{271}{164}a+\frac{31}{82}$, $\frac{75}{287}a^{15}+\frac{44}{287}a^{14}-\frac{583}{287}a^{13}-\frac{619}{574}a^{12}+\frac{9213}{1148}a^{11}+\frac{4521}{1148}a^{10}-\frac{20499}{1148}a^{9}-\frac{8915}{1148}a^{8}+\frac{7358}{287}a^{7}+\frac{2897}{287}a^{6}-\frac{6570}{287}a^{5}-\frac{4437}{574}a^{4}+\frac{11507}{1148}a^{3}+\frac{2507}{1148}a^{2}+\frac{179}{1148}a+\frac{591}{1148}$, $\frac{113}{1148}a^{15}+\frac{1}{41}a^{14}-\frac{909}{1148}a^{13}-\frac{15}{82}a^{12}+\frac{1837}{574}a^{11}+\frac{113}{164}a^{10}-\frac{4195}{574}a^{9}-\frac{211}{164}a^{8}+\frac{12169}{1148}a^{7}+\frac{50}{41}a^{6}-\frac{10645}{1148}a^{5}-\frac{3}{82}a^{4}+\frac{870}{287}a^{3}-\frac{121}{164}a^{2}+\frac{538}{287}a+\frac{115}{164}$, $\frac{111}{1148}a^{15}-\frac{49}{164}a^{14}-\frac{771}{1148}a^{13}+\frac{97}{41}a^{12}+\frac{1327}{574}a^{11}-\frac{769}{82}a^{10}-\frac{2287}{574}a^{9}+\frac{3415}{164}a^{8}+\frac{4197}{1148}a^{7}-\frac{4787}{164}a^{6}-\frac{597}{1148}a^{5}+\frac{1028}{41}a^{4}-\frac{1971}{574}a^{3}-\frac{781}{82}a^{2}+\frac{1857}{574}a-\frac{271}{164}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 658.644099006 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 658.644099006 \cdot 1}{8\cdot\sqrt{288230376151711744}}\cr\approx \mathstrut & 0.372503099559 \end{aligned}\] (assuming GRH)
Galois group
$\SD_{16}$ (as 16T12):
A solvable group of order 16 |
The 7 conjugacy class representatives for $QD_{16}$ |
Character table for $QD_{16}$ |
Intermediate fields
\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\), 4.2.1024.1 x2, 4.0.512.1 x2, 8.0.4194304.1, 8.2.268435456.2 x4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | 8.2.268435456.2 |
Minimal sibling: | 8.2.268435456.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.1.0.1}{1} }^{16}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.58.2 | $x^{16} + 12 x^{14} + 8 x^{13} + 8 x^{11} + 8 x^{10} + 6 x^{8} + 16 x^{7} + 28 x^{4} + 2$ | $16$ | $1$ | $58$ | $QD_{16}$ | $[2, 3, 7/2, 9/2]$ |