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Properties

Label 16.0.68823182750...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{4}\cdot 17^{8}\cdot 97^{2}$
Root discriminant $30.89$
Ramified primes $2, 5, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1912, 4280, -2542, -8860, 10837, 3078, -9057, 3358, 2211, -1644, 44, 192, 39, -18, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 - 18*x^13 + 39*x^12 + 192*x^11 + 44*x^10 - 1644*x^9 + 2211*x^8 + 3358*x^7 - 9057*x^6 + 3078*x^5 + 10837*x^4 - 8860*x^3 - 2542*x^2 + 4280*x + 1912)
 
gp: K = bnfinit(x^16 - 2*x^15 - 5*x^14 - 18*x^13 + 39*x^12 + 192*x^11 + 44*x^10 - 1644*x^9 + 2211*x^8 + 3358*x^7 - 9057*x^6 + 3078*x^5 + 10837*x^4 - 8860*x^3 - 2542*x^2 + 4280*x + 1912, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 5 x^{14} - 18 x^{13} + 39 x^{12} + 192 x^{11} + 44 x^{10} - 1644 x^{9} + 2211 x^{8} + 3358 x^{7} - 9057 x^{6} + 3078 x^{5} + 10837 x^{4} - 8860 x^{3} - 2542 x^{2} + 4280 x + 1912 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(688231827503778365440000=2^{24}\cdot 5^{4}\cdot 17^{8}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.89$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 17, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{9} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{7}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{10} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{7}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{5}{16} a^{7} + \frac{3}{8} a^{6} + \frac{1}{8} a^{5} + \frac{3}{8} a^{4} - \frac{7}{16} a^{3} - \frac{3}{8} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{832} a^{14} - \frac{1}{52} a^{13} + \frac{1}{104} a^{12} + \frac{11}{416} a^{11} - \frac{85}{832} a^{10} + \frac{17}{208} a^{9} + \frac{179}{832} a^{8} + \frac{59}{416} a^{7} - \frac{197}{416} a^{6} - \frac{5}{13} a^{5} - \frac{67}{832} a^{4} + \frac{3}{13} a^{3} + \frac{119}{416} a^{2} - \frac{23}{104} a - \frac{7}{104}$, $\frac{1}{657699017597262169425728} a^{15} + \frac{38926507412511583069}{657699017597262169425728} a^{14} - \frac{72998163098903432783}{20553094299914442794554} a^{13} - \frac{8533813360625618341343}{328849508798631084712864} a^{12} + \frac{19442919277164964792633}{657699017597262169425728} a^{11} - \frac{78104948490319070895273}{657699017597262169425728} a^{10} + \frac{62743238204974148972511}{657699017597262169425728} a^{9} + \frac{128862961283376767941677}{657699017597262169425728} a^{8} - \frac{27015431418196128516207}{164424754399315542356432} a^{7} - \frac{95305815782538241609083}{328849508798631084712864} a^{6} - \frac{185619468011301476688915}{657699017597262169425728} a^{5} - \frac{88149353387395501197779}{657699017597262169425728} a^{4} + \frac{49746564634794947933535}{328849508798631084712864} a^{3} - \frac{92122719286321008265085}{328849508798631084712864} a^{2} - \frac{17274255010487046577145}{41106188599828885589108} a - \frac{490802741457601005}{26460372449197866488}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2910940.1755 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), 4.0.2312.1 x2, 4.0.1088.2 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.342102016.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$