Normalized defining polynomial
\( x^{16} + 42x^{12} + 83x^{8} + 42x^{4} + 1 \)
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: | \(6871947673600000000\) \(\medspace = 2^{44}\cdot 5^{8}\) | 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: | \(15.04\) | 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: | $2^{11/4}5^{1/2}\approx 15.042412372345574$ | ||
Ramified primes: | \(2\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{8}-\frac{1}{2}a^{4}+\frac{1}{6}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}+\frac{1}{6}a$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{156}a^{12}-\frac{1}{12}a^{10}+\frac{1}{39}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{14}{39}a^{4}-\frac{1}{2}a^{3}-\frac{1}{12}a^{2}-\frac{1}{2}a+\frac{25}{156}$, $\frac{1}{156}a^{13}-\frac{1}{12}a^{11}+\frac{1}{39}a^{9}-\frac{1}{4}a^{7}+\frac{11}{78}a^{5}-\frac{1}{2}a^{4}+\frac{5}{12}a^{3}-\frac{1}{2}a^{2}-\frac{53}{156}a-\frac{1}{2}$, $\frac{1}{156}a^{14}-\frac{3}{52}a^{10}-\frac{1}{12}a^{8}-\frac{17}{156}a^{6}+\frac{1}{4}a^{4}-\frac{11}{26}a^{2}-\frac{1}{12}$, $\frac{1}{156}a^{15}-\frac{3}{52}a^{11}-\frac{1}{12}a^{9}-\frac{17}{156}a^{7}+\frac{1}{4}a^{5}-\frac{11}{26}a^{3}-\frac{1}{12}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{7}{78} a^{14} + \frac{7}{52} a^{12} + \frac{589}{156} a^{10} + \frac{72}{13} a^{8} + \frac{1205}{156} a^{6} + \frac{84}{13} a^{4} + \frac{727}{156} a^{2} + \frac{45}{52} \) (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: | $\frac{5}{78}a^{13}+\frac{215}{78}a^{9}+\frac{617}{78}a^{5}+\frac{160}{39}a$, $\frac{7}{156}a^{14}-\frac{1}{39}a^{12}+\frac{275}{156}a^{10}-\frac{53}{52}a^{8}-\frac{197}{156}a^{6}+\frac{29}{156}a^{4}-\frac{109}{39}a^{2}+\frac{23}{52}$, $\frac{31}{156}a^{15}-\frac{1}{6}a^{14}-\frac{5}{156}a^{13}-\frac{7}{52}a^{12}+\frac{647}{78}a^{11}-\frac{83}{12}a^{10}-\frac{215}{156}a^{9}-\frac{72}{13}a^{8}+\frac{1121}{78}a^{7}-\frac{125}{12}a^{6}-\frac{617}{156}a^{5}-\frac{84}{13}a^{4}+\frac{853}{156}a^{3}-\frac{41}{12}a^{2}-\frac{80}{39}a-\frac{71}{52}$, $\frac{31}{156}a^{15}-\frac{1}{6}a^{14}-\frac{5}{156}a^{13}+\frac{7}{52}a^{12}+\frac{647}{78}a^{11}-\frac{83}{12}a^{10}-\frac{215}{156}a^{9}+\frac{72}{13}a^{8}+\frac{1121}{78}a^{7}-\frac{125}{12}a^{6}-\frac{617}{156}a^{5}+\frac{84}{13}a^{4}+\frac{853}{156}a^{3}-\frac{41}{12}a^{2}-\frac{80}{39}a+\frac{19}{52}$, $\frac{4}{13}a^{15}+\frac{15}{52}a^{13}+\frac{1999}{156}a^{11}+\frac{461}{39}a^{9}+\frac{1093}{52}a^{7}+\frac{154}{13}a^{5}+\frac{979}{156}a^{3}-\frac{97}{156}a$, $\frac{10}{39}a^{15}-\frac{17}{39}a^{14}+\frac{7}{26}a^{13}-\frac{11}{52}a^{12}+\frac{139}{13}a^{11}-\frac{2807}{156}a^{10}+\frac{144}{13}a^{9}-\frac{677}{78}a^{8}+\frac{1415}{78}a^{7}-\frac{3641}{156}a^{6}+\frac{168}{13}a^{5}-\frac{119}{13}a^{4}+\frac{105}{13}a^{3}-\frac{1037}{156}a^{2}+\frac{42}{13}a-\frac{97}{156}$, $\frac{1}{13}a^{15}-\frac{1}{6}a^{14}+\frac{3}{52}a^{12}+\frac{245}{78}a^{11}-\frac{83}{12}a^{10}+\frac{187}{78}a^{8}+\frac{35}{13}a^{7}-\frac{125}{12}a^{6}+\frac{49}{13}a^{4}-\frac{97}{78}a^{3}-\frac{35}{12}a^{2}-\frac{1}{2}a+\frac{173}{156}$ | 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: | \( 2592.42198789 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2592.42198789 \cdot 1}{8\cdot\sqrt{6871947673600000000}}\cr\approx \mathstrut & 0.300271929185 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 16T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.4.655360000.2, 8.0.2621440000.12, 8.0.163840000.3, 8.0.2621440000.8 |
Minimal sibling: | 8.0.163840000.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.1.0.1}{1} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.44.2 | $x^{16} + 8 x^{15} + 20 x^{14} + 16 x^{13} + 16 x^{12} + 8 x^{11} + 72 x^{10} + 136 x^{9} + 136 x^{8} + 160 x^{7} + 136 x^{6} + 208 x^{5} + 240 x^{4} + 208 x^{3} + 160 x^{2} + 48 x + 36$ | $8$ | $2$ | $44$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ |
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |