↑

Properties

Label 16.0.17669691567...761.20
Degree $16$
Signature $[0, 8]$
Discriminant $23^{8}\cdot 41^{12}$
Root discriminant $77.71$
Ramified primes $23, 41$
Class number $432$ (GRH)
Class group $[6, 6, 12]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34932400, 10864280, 4560832, -9955312, 4034535, -1078216, 300929, 30444, 2030, -22602, 11738, -1310, -160, -14, 31, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 31*x^14 - 14*x^13 - 160*x^12 - 1310*x^11 + 11738*x^10 - 22602*x^9 + 2030*x^8 + 30444*x^7 + 300929*x^6 - 1078216*x^5 + 4034535*x^4 - 9955312*x^3 + 4560832*x^2 + 10864280*x + 34932400)
 
gp: K = bnfinit(x^16 - 6*x^15 + 31*x^14 - 14*x^13 - 160*x^12 - 1310*x^11 + 11738*x^10 - 22602*x^9 + 2030*x^8 + 30444*x^7 + 300929*x^6 - 1078216*x^5 + 4034535*x^4 - 9955312*x^3 + 4560832*x^2 + 10864280*x + 34932400, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 31 x^{14} - 14 x^{13} - 160 x^{12} - 1310 x^{11} + 11738 x^{10} - 22602 x^{9} + 2030 x^{8} + 30444 x^{7} + 300929 x^{6} - 1078216 x^{5} + 4034535 x^{4} - 9955312 x^{3} + 4560832 x^{2} + 10864280 x + 34932400 \)

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:  \(1766969156799962667099298073761=23^{8}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.71$
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:  $23, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{3}{10} a^{5} - \frac{1}{10} a^{2} + \frac{1}{5} a$, $\frac{1}{60} a^{12} + \frac{1}{15} a^{10} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{3} a^{4} - \frac{1}{10} a^{3} + \frac{11}{60} a^{2} + \frac{1}{5} a + \frac{1}{3}$, $\frac{1}{60} a^{13} - \frac{1}{30} a^{11} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{11}{30} a^{5} - \frac{1}{10} a^{4} + \frac{11}{60} a^{3} - \frac{1}{5} a^{2} + \frac{2}{15} a$, $\frac{1}{1106760} a^{14} - \frac{351}{73784} a^{13} + \frac{1021}{184460} a^{12} - \frac{2839}{92230} a^{11} - \frac{17677}{276690} a^{10} - \frac{12173}{184460} a^{9} + \frac{81}{802} a^{8} - \frac{23267}{184460} a^{7} - \frac{43303}{276690} a^{6} - \frac{1922}{46115} a^{5} + \frac{6767}{368920} a^{4} + \frac{16263}{73784} a^{3} - \frac{6835}{18446} a^{2} - \frac{776}{2005} a - \frac{421}{1203}$, $\frac{1}{24953327709720219139646762541744402499076703480} a^{15} - \frac{1291591898516305174668363890309045427437}{3119165963715027392455845317718050312384587935} a^{14} + \frac{5849287937236638073214594229374891100132011}{1084927291726966049549859240945408804307682760} a^{13} + \frac{3061306048720576711251442697790316841924027}{623833192743005478491169063543610062476917587} a^{12} - \frac{83107960866586303019516104650683687666739969}{2079443975810018261637230211812033541589725290} a^{11} - \frac{594985287201188528030219843528048132230081153}{4158887951620036523274460423624067083179450580} a^{10} + \frac{83290939723922169856314562038061829373572379}{831777590324007304654892084724813416635890116} a^{9} + \frac{1019873784850641348876491943901767776546939531}{4158887951620036523274460423624067083179450580} a^{8} + \frac{672864277241092343976662486375139321462203419}{12476663854860109569823381270872201249538351740} a^{7} - \frac{657403289938728857431375053693356247161783632}{3119165963715027392455845317718050312384587935} a^{6} - \frac{178980518604700404962310703067508639355038227}{4990665541944043827929352508348880499815340696} a^{5} + \frac{3744712368732495874486202182838593412833996541}{12476663854860109569823381270872201249538351740} a^{4} + \frac{2971913337858020723042301303835399685400347453}{24953327709720219139646762541744402499076703480} a^{3} + \frac{4550631059084605647853487270662859860124344301}{12476663854860109569823381270872201249538351740} a^{2} + \frac{21702042500539102203000049593898627427608324}{45205303821956918731244135039392033512820115} a + \frac{523583461138142021989345460747809322837582}{9041060764391383746248827007878406702564023}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{12}$, which has order $432$ (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:  \( 1119246.56118 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-943}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-23}, \sqrt{41})\), 4.2.38663.1 x2, 4.0.21689.1 x2, 8.0.790763784001.2, 8.2.57794518300247.1 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
$41$41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$