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Properties

Label 16.8.14014641336...5089.3
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 41^{15}$
Root discriminant $765.88$
Ramified primes $37, 41$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![77696000, 115686400, -1266360960, 93759936, 5420592940, -6408866240, 1858998781, 285518672, -131779642, 41196890, -9169571, 507300, -84459, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 84459*x^12 + 507300*x^11 - 9169571*x^10 + 41196890*x^9 - 131779642*x^8 + 285518672*x^7 + 1858998781*x^6 - 6408866240*x^5 + 5420592940*x^4 + 93759936*x^3 - 1266360960*x^2 + 115686400*x + 77696000)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 84459*x^12 + 507300*x^11 - 9169571*x^10 + 41196890*x^9 - 131779642*x^8 + 285518672*x^7 + 1858998781*x^6 - 6408866240*x^5 + 5420592940*x^4 + 93759936*x^3 - 1266360960*x^2 + 115686400*x + 77696000, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 84459 x^{12} + 507300 x^{11} - 9169571 x^{10} + 41196890 x^{9} - 131779642 x^{8} + 285518672 x^{7} + 1858998781 x^{6} - 6408866240 x^{5} + 5420592940 x^{4} + 93759936 x^{3} - 1266360960 x^{2} + 115686400 x + 77696000 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14014641336002292789493279203934745080101945089=37^{14}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $765.88$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{40} a^{7} - \frac{1}{40} a^{6} - \frac{1}{5} a^{5} + \frac{1}{8} a^{4} - \frac{1}{40} a^{3} + \frac{2}{5} a^{2} - \frac{3}{10} a$, $\frac{1}{1480} a^{8} - \frac{1}{370} a^{7} - \frac{3}{148} a^{6} + \frac{13}{185} a^{5} + \frac{97}{740} a^{4} - \frac{283}{740} a^{3} - \frac{5}{296} a^{2} + \frac{163}{740} a + \frac{4}{37}$, $\frac{1}{1480} a^{9} - \frac{9}{1480} a^{7} - \frac{53}{1480} a^{6} + \frac{157}{740} a^{5} - \frac{69}{296} a^{4} + \frac{317}{740} a^{3} - \frac{331}{740} a^{2} + \frac{7}{37} a + \frac{16}{37}$, $\frac{1}{59200} a^{10} - \frac{1}{11840} a^{9} - \frac{1}{3700} a^{8} + \frac{47}{29600} a^{7} + \frac{55}{1184} a^{6} - \frac{43}{296} a^{5} - \frac{3439}{59200} a^{4} + \frac{21331}{59200} a^{3} + \frac{1983}{7400} a^{2} - \frac{1399}{2960} a + \frac{41}{148}$, $\frac{1}{296000} a^{11} + \frac{1}{148000} a^{10} + \frac{29}{296000} a^{9} + \frac{11}{148000} a^{8} - \frac{27}{18500} a^{7} + \frac{817}{29600} a^{6} - \frac{10679}{296000} a^{5} + \frac{10909}{148000} a^{4} + \frac{18621}{296000} a^{3} - \frac{34163}{74000} a^{2} - \frac{493}{2960} a + \frac{283}{740}$, $\frac{1}{47360000} a^{12} - \frac{3}{23680000} a^{11} - \frac{87}{47360000} a^{10} + \frac{49}{4736000} a^{9} - \frac{901}{5920000} a^{8} + \frac{349}{640000} a^{7} - \frac{2365239}{47360000} a^{6} + \frac{140149}{947200} a^{5} + \frac{157221}{1280000} a^{4} - \frac{145527}{296000} a^{3} + \frac{731679}{11840000} a^{2} + \frac{12349}{59200} a - \frac{13}{800}$, $\frac{1}{47360000} a^{13} + \frac{1}{1280000} a^{11} + \frac{9}{1480000} a^{10} + \frac{93}{11840000} a^{9} - \frac{6951}{23680000} a^{8} + \frac{88597}{47360000} a^{7} - \frac{3927}{80000} a^{6} + \frac{9449237}{47360000} a^{5} - \frac{4285189}{23680000} a^{4} + \frac{79947}{320000} a^{3} - \frac{2707103}{5920000} a^{2} - \frac{2081}{7400} a + \frac{4217}{14800}$, $\frac{1}{24279732373055897600000} a^{14} - \frac{7}{24279732373055897600000} a^{13} + \frac{189179887504219}{24279732373055897600000} a^{12} - \frac{1135079325025223}{24279732373055897600000} a^{11} - \frac{10073382571743299}{12139866186527948800000} a^{10} + \frac{55569359765082017}{12139866186527948800000} a^{9} - \frac{1239048617011442533}{4855946474611179520000} a^{8} + \frac{24101654150472589289}{24279732373055897600000} a^{7} - \frac{1379882854098193817973}{24279732373055897600000} a^{6} + \frac{4055772041262529357503}{24279732373055897600000} a^{5} - \frac{724008476219077847083}{3034966546631987200000} a^{4} + \frac{1213101406716975002921}{6069933093263974400000} a^{3} + \frac{27961324341263627807}{60699330932639744000} a^{2} + \frac{7088111984469043231}{15174832733159936000} a + \frac{14615500918034269}{151748327331599360}$, $\frac{1}{121398661865279488000000} a^{15} + \frac{70184315551633}{12139866186527948800000} a^{13} + \frac{70184315551647}{12139866186527948800000} a^{12} - \frac{94225881992231799}{121398661865279488000000} a^{11} - \frac{11361945670961149}{15174832733159936000000} a^{10} - \frac{40164563027674244907}{121398661865279488000000} a^{9} + \frac{16864477284744539917}{60699330932639744000000} a^{8} - \frac{108426507169135157007}{12139866186527948800000} a^{7} + \frac{808057290713995880003}{30349665466319872000000} a^{6} + \frac{28433124773220115662177}{121398661865279488000000} a^{5} + \frac{2359152897362313017519}{30349665466319872000000} a^{4} - \frac{2012632537233397791813}{30349665466319872000000} a^{3} - \frac{732983112907993065911}{1517483273315993600000} a^{2} - \frac{15741940037984198803}{75874163665799680000} a + \frac{855872816091566679}{3793708183289984000}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  $11$
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:  \( 1106489458250000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.94352849.1, 8.8.499686183762100623329.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{12}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{12}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
41Data not computed