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Properties

Label 16.0.16994144084...8576.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 223^{4}$
Root discriminant $18.38$
Ramified primes $2, 223$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^2:S_4:C_2$ (as 16T724)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![68, 80, 152, -128, -32, -104, 144, -336, 398, -288, 192, -92, 44, -24, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 24*x^13 + 44*x^12 - 92*x^11 + 192*x^10 - 288*x^9 + 398*x^8 - 336*x^7 + 144*x^6 - 104*x^5 - 32*x^4 - 128*x^3 + 152*x^2 + 80*x + 68)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 24*x^13 + 44*x^12 - 92*x^11 + 192*x^10 - 288*x^9 + 398*x^8 - 336*x^7 + 144*x^6 - 104*x^5 - 32*x^4 - 128*x^3 + 152*x^2 + 80*x + 68, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 24 x^{13} + 44 x^{12} - 92 x^{11} + 192 x^{10} - 288 x^{9} + 398 x^{8} - 336 x^{7} + 144 x^{6} - 104 x^{5} - 32 x^{4} - 128 x^{3} + 152 x^{2} + 80 x + 68 \)

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:  \(169941440847545368576=2^{36}\cdot 223^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.38$
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:  $2, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{26} a^{13} - \frac{3}{26} a^{12} - \frac{1}{13} a^{11} + \frac{1}{13} a^{10} - \frac{2}{13} a^{9} + \frac{1}{26} a^{8} + \frac{5}{13} a^{7} + \frac{4}{13} a^{6} - \frac{4}{13} a^{5} + \frac{5}{13} a^{4} + \frac{3}{13} a^{2} + \frac{5}{13} a + \frac{5}{13}$, $\frac{1}{338} a^{14} + \frac{1}{338} a^{13} + \frac{51}{338} a^{12} - \frac{71}{338} a^{11} + \frac{2}{169} a^{10} + \frac{38}{169} a^{9} + \frac{20}{169} a^{8} - \frac{28}{169} a^{7} - \frac{66}{169} a^{6} + \frac{15}{169} a^{5} - \frac{32}{169} a^{4} - \frac{10}{169} a^{3} - \frac{35}{169} a^{2} + \frac{77}{169} a + \frac{46}{169}$, $\frac{1}{291429541026} a^{15} + \frac{39890854}{48571590171} a^{14} - \frac{2571783419}{291429541026} a^{13} + \frac{7505413664}{145714770513} a^{12} + \frac{2439866811}{32381060114} a^{11} - \frac{1906100507}{291429541026} a^{10} + \frac{5601616273}{291429541026} a^{9} + \frac{891813103}{22417657002} a^{8} - \frac{15153702307}{48571590171} a^{7} + \frac{7006034354}{16190530057} a^{6} + \frac{5636030419}{16190530057} a^{5} + \frac{54943640588}{145714770513} a^{4} + \frac{30746194924}{145714770513} a^{3} - \frac{7387920353}{48571590171} a^{2} - \frac{63665754734}{145714770513} a + \frac{2193612298}{8571457089}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{29520497}{1502214129} a^{15} + \frac{24831134}{500738043} a^{14} - \frac{156251399}{1502214129} a^{13} + \frac{380678254}{1502214129} a^{12} - \frac{58854436}{166912681} a^{11} + \frac{2837057471}{3004428258} a^{10} - \frac{2673037793}{1502214129} a^{9} + \frac{203905444}{115554933} a^{8} - \frac{1271881913}{500738043} a^{7} - \frac{90758167}{166912681} a^{6} + \frac{123224614}{166912681} a^{5} + \frac{47368846}{1502214129} a^{4} + \frac{1272638045}{1502214129} a^{3} + \frac{1784058437}{500738043} a^{2} + \frac{834872912}{1502214129} a + \frac{6149969}{88365537} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 16318.239899 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:S_4:C_2$ (as 16T724):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 28 conjugacy class representatives for $C_2\times C_2^2:S_4:C_2$
Character table for $C_2\times C_2^2:S_4:C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), 8.0.3259039744.8

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
223Data not computed