Normalized defining polynomial
\( x^{16} - 24x^{14} + 220x^{12} - 984x^{10} + 2262x^{8} - 2568x^{6} + 1180x^{4} - 72x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1495873430980877352960000\) \(\medspace = 2^{52}\cdot 3^{12}\cdot 5^{4}\) | 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: | \(32.43\) | 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: | $2^{13/4}3^{3/4}5^{1/2}\approx 48.49237211223896$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{3}{8}a^{2}-\frac{1}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{8}-\frac{1}{8}a^{4}+\frac{1}{8}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{3}{16}a^{5}-\frac{3}{16}a^{4}+\frac{5}{16}a-\frac{5}{16}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{12}-\frac{1}{16}a^{10}+\frac{1}{16}a^{8}+\frac{3}{16}a^{6}-\frac{3}{16}a^{4}+\frac{5}{16}a^{2}-\frac{5}{16}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}+\frac{1}{16}a^{8}-\frac{1}{16}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{16}a^{4}+\frac{1}{16}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{16}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{1}{8}a^{12}-\frac{11}{4}a^{10}+\frac{175}{8}a^{8}-77a^{6}+\frac{923}{8}a^{4}-\frac{233}{4}a^{2}+\frac{5}{8}$, $\frac{7}{16}a^{15}-\frac{167}{16}a^{13}+\frac{1517}{16}a^{11}-\frac{6693}{16}a^{9}+\frac{15077}{16}a^{7}-\frac{16621}{16}a^{5}+\frac{7247}{16}a^{3}-\frac{303}{16}a$, $\frac{7}{16}a^{15}-\frac{165}{16}a^{13}+\frac{1473}{16}a^{11}-\frac{6339}{16}a^{9}+\frac{13781}{16}a^{7}-\frac{14471}{16}a^{5}+\frac{5851}{16}a^{3}-\frac{89}{16}a$, $\frac{3}{4}a^{15}-\frac{145}{8}a^{13}+\frac{671}{4}a^{11}-\frac{6079}{8}a^{9}+\frac{7093}{4}a^{7}-\frac{16299}{8}a^{5}+\frac{3701}{4}a^{3}-\frac{269}{8}a$, $\frac{29}{16}a^{15}-\frac{695}{16}a^{13}+\frac{6357}{16}a^{11}-\frac{28335}{16}a^{9}+\frac{64739}{16}a^{7}-\frac{72569}{16}a^{5}+\frac{32123}{16}a^{3}-\frac{1169}{16}a$, $\frac{1}{16}a^{15}+\frac{1}{16}a^{14}-\frac{13}{8}a^{13}-\frac{13}{8}a^{12}+\frac{263}{16}a^{11}+\frac{263}{16}a^{10}-\frac{659}{8}a^{9}-\frac{659}{8}a^{8}+\frac{3419}{16}a^{7}+\frac{3419}{16}a^{6}-\frac{2155}{8}a^{5}-\frac{2155}{8}a^{4}+\frac{2085}{16}a^{3}+\frac{2085}{16}a^{2}-\frac{29}{8}a-\frac{29}{8}$, $\frac{13}{16}a^{15}+\frac{1}{16}a^{14}-\frac{311}{16}a^{13}-\frac{23}{16}a^{12}+\frac{2837}{16}a^{11}+\frac{197}{16}a^{10}-\frac{12595}{16}a^{9}-\frac{791}{16}a^{8}+\frac{28615}{16}a^{7}+\frac{1539}{16}a^{6}-\frac{31845}{16}a^{5}-\frac{1389}{16}a^{4}+\frac{13951}{16}a^{3}+\frac{463}{16}a^{2}-\frac{457}{16}a+\frac{3}{16}$, $\frac{1}{2}a^{15}+\frac{1}{16}a^{14}-\frac{193}{16}a^{13}-\frac{11}{8}a^{12}+\frac{891}{8}a^{11}+\frac{175}{16}a^{10}-\frac{8045}{16}a^{9}-\frac{307}{8}a^{8}+\frac{4669}{4}a^{7}+\frac{891}{16}a^{6}-\frac{21255}{16}a^{5}-\frac{161}{8}a^{4}+\frac{4715}{8}a^{3}-\frac{163}{16}a^{2}-\frac{235}{16}a+\frac{7}{8}$, $\frac{1}{2}a^{15}-\frac{1}{16}a^{14}-\frac{193}{16}a^{13}+\frac{11}{8}a^{12}+\frac{891}{8}a^{11}-\frac{175}{16}a^{10}-\frac{8045}{16}a^{9}+\frac{307}{8}a^{8}+\frac{4669}{4}a^{7}-\frac{891}{16}a^{6}-\frac{21255}{16}a^{5}+\frac{161}{8}a^{4}+\frac{4715}{8}a^{3}+\frac{163}{16}a^{2}-\frac{235}{16}a-\frac{7}{8}$, $\frac{17}{16}a^{15}-\frac{101}{4}a^{13}+\frac{3}{16}a^{12}+\frac{3651}{16}a^{11}-\frac{33}{8}a^{10}-\frac{4001}{4}a^{9}+\frac{527}{16}a^{8}+\frac{35783}{16}a^{7}-\frac{235}{2}a^{6}-2450a^{5}+\frac{2921}{16}a^{4}+\frac{17077}{16}a^{3}-\frac{811}{8}a^{2}-44a+\frac{77}{16}$, $\frac{17}{16}a^{15}-\frac{1}{8}a^{14}-\frac{101}{4}a^{13}+\frac{47}{16}a^{12}+\frac{3651}{16}a^{11}-\frac{209}{8}a^{10}-\frac{4001}{4}a^{9}+\frac{1793}{16}a^{8}+\frac{35783}{16}a^{7}-\frac{1953}{8}a^{6}-2450a^{5}+\frac{4185}{16}a^{4}+\frac{17077}{16}a^{3}-\frac{897}{8}a^{2}-44a+\frac{71}{16}$, $\frac{7}{16}a^{15}+\frac{1}{8}a^{14}-\frac{87}{8}a^{13}-\frac{47}{16}a^{12}+\frac{1671}{16}a^{11}+\frac{209}{8}a^{10}-\frac{1981}{4}a^{9}-\frac{1793}{16}a^{8}+\frac{19485}{16}a^{7}+\frac{1953}{8}a^{6}-\frac{11765}{8}a^{5}-\frac{4185}{16}a^{4}+\frac{11117}{16}a^{3}+\frac{897}{8}a^{2}-\frac{109}{4}a-\frac{71}{16}$, $\frac{9}{8}a^{15}-\frac{1}{8}a^{14}-\frac{433}{16}a^{13}+\frac{49}{16}a^{12}+\frac{995}{4}a^{11}-\frac{231}{8}a^{10}-\frac{17845}{16}a^{9}+\frac{2143}{16}a^{8}+\frac{20507}{8}a^{7}-\frac{2569}{8}a^{6}-\frac{46067}{16}a^{5}+\frac{6031}{16}a^{4}+\frac{5023}{4}a^{3}-\frac{1363}{8}a^{2}-\frac{463}{16}a+\frac{65}{16}$, $\frac{11}{16}a^{15}+\frac{1}{16}a^{14}-\frac{265}{16}a^{13}-\frac{25}{16}a^{12}+\frac{2441}{16}a^{11}+\frac{241}{16}a^{10}-\frac{10979}{16}a^{9}-\frac{1141}{16}a^{8}+\frac{25357}{16}a^{7}+\frac{2771}{16}a^{6}-\frac{28735}{16}a^{5}-\frac{3243}{16}a^{4}+\frac{12863}{16}a^{3}+\frac{1459}{16}a^{2}-\frac{533}{16}a-\frac{79}{16}$, $\frac{3}{16}a^{15}-\frac{1}{4}a^{14}-\frac{73}{16}a^{13}+\frac{23}{4}a^{12}+\frac{681}{16}a^{11}-\frac{397}{8}a^{10}-\frac{3107}{16}a^{9}+\frac{1633}{8}a^{8}+\frac{7261}{16}a^{7}-\frac{1675}{4}a^{6}-\frac{8191}{16}a^{5}+\frac{1651}{4}a^{4}+\frac{3415}{16}a^{3}-\frac{1267}{8}a^{2}+\frac{75}{16}a+\frac{27}{8}$ (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: | \( 14732523.2621 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 14732523.2621 \cdot 1}{2\cdot\sqrt{1495873430980877352960000}}\cr\approx \mathstrut & 0.394711378550 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2:Q_8$ (as 16T31):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_2^2:Q_8$ |
Character table for $C_2^2:Q_8$ |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.92160.1, \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.92160.2, 8.8.33973862400.1, 8.8.19110297600.1, 8.8.12230590464.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 sibling: | deg 16 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.52.15 | $x^{16} + 8 x^{15} + 12 x^{14} + 8 x^{11} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 30$ | $16$ | $1$ | $52$ | 16T31 | $[2, 3, 3, 4]^{2}$ |
\(3\) | 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
\(5\) | 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |