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Properties

Label 16.8.20027199604...9569.2
Degree $16$
Signature $[8, 4]$
Discriminant $71^{12}\cdot 73^{14}$
Root discriminant $1044.36$
Ramified primes $71, 73$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3837587340384256, 962164281331200, -434316312830208, -259884347484288, -55216022400064, 1150878405120, 279834449792, 543896, 8179623956, -123129514, -11510253, 803174, -205185, 562, 25, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 25*x^14 + 562*x^13 - 205185*x^12 + 803174*x^11 - 11510253*x^10 - 123129514*x^9 + 8179623956*x^8 + 543896*x^7 + 279834449792*x^6 + 1150878405120*x^5 - 55216022400064*x^4 - 259884347484288*x^3 - 434316312830208*x^2 + 962164281331200*x + 3837587340384256)
 
gp: K = bnfinit(x^16 - 6*x^15 + 25*x^14 + 562*x^13 - 205185*x^12 + 803174*x^11 - 11510253*x^10 - 123129514*x^9 + 8179623956*x^8 + 543896*x^7 + 279834449792*x^6 + 1150878405120*x^5 - 55216022400064*x^4 - 259884347484288*x^3 - 434316312830208*x^2 + 962164281331200*x + 3837587340384256, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 25 x^{14} + 562 x^{13} - 205185 x^{12} + 803174 x^{11} - 11510253 x^{10} - 123129514 x^{9} + 8179623956 x^{8} + 543896 x^{7} + 279834449792 x^{6} + 1150878405120 x^{5} - 55216022400064 x^{4} - 259884347484288 x^{3} - 434316312830208 x^{2} + 962164281331200 x + 3837587340384256 \)

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:  \(2002719960469295074007887826028741898882244399569=71^{12}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1044.36$
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:  $71, 73$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4}$, $\frac{1}{32} a^{7} - \frac{1}{32} a^{6} - \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{8} - \frac{1}{64} a^{7} - \frac{1}{64} a^{6} + \frac{1}{64} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{64} a^{9} - \frac{1}{32} a^{6} - \frac{1}{64} a^{5} - \frac{3}{32} a^{4} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{128} a^{10} - \frac{1}{64} a^{7} - \frac{1}{128} a^{6} - \frac{3}{64} a^{5} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{256} a^{11} - \frac{1}{128} a^{8} - \frac{1}{256} a^{7} - \frac{3}{128} a^{6} - \frac{1}{32} a^{4} + \frac{1}{16} a^{3}$, $\frac{1}{1024} a^{12} - \frac{1}{1024} a^{11} - \frac{1}{512} a^{10} - \frac{3}{512} a^{9} - \frac{7}{1024} a^{8} + \frac{7}{1024} a^{7} + \frac{7}{128} a^{6} - \frac{1}{32} a^{5} + \frac{1}{64} a^{4} + \frac{3}{32} a^{3} - \frac{1}{16} a^{2} - \frac{5}{16} a + \frac{1}{4}$, $\frac{1}{335872} a^{13} + \frac{45}{335872} a^{12} + \frac{151}{83968} a^{11} - \frac{365}{167936} a^{10} + \frac{1237}{335872} a^{9} + \frac{781}{335872} a^{8} - \frac{2433}{167936} a^{7} - \frac{445}{10496} a^{6} + \frac{777}{20992} a^{5} - \frac{309}{5248} a^{4} - \frac{43}{656} a^{3} + \frac{277}{5248} a^{2} - \frac{371}{2624} a + \frac{165}{656}$, $\frac{1}{599195648} a^{14} + \frac{139}{599195648} a^{13} + \frac{136897}{299597824} a^{12} + \frac{472791}{299597824} a^{11} + \frac{672585}{599195648} a^{10} - \frac{730493}{599195648} a^{9} + \frac{487817}{149798912} a^{8} - \frac{41815}{149798912} a^{7} + \frac{501285}{37449728} a^{6} + \frac{395717}{18724864} a^{5} - \frac{349583}{4681216} a^{4} - \frac{1984971}{9362432} a^{3} + \frac{102933}{585152} a^{2} + \frac{292525}{2340608} a + \frac{180283}{585152}$, $\frac{1}{14120155289281468310898468098639238211024163024706626179379996516086182525556193169833984} a^{15} - \frac{5660557186021427212480684258073650597646306373553827516831194315578807681019859}{7060077644640734155449234049319619105512081512353313089689998258043091262778096584916992} a^{14} - \frac{10319610228736124477388268962219608201035578314330086732806715305774310432182187073}{14120155289281468310898468098639238211024163024706626179379996516086182525556193169833984} a^{13} + \frac{781678544034085344245351490516666599742262210896382108455349460683097203719248000497}{3530038822320367077724617024659809552756040756176656544844999129021545631389048292458496} a^{12} + \frac{20223498201072767671451617291756673924392382925663154688497675962695637737147661749659}{14120155289281468310898468098639238211024163024706626179379996516086182525556193169833984} a^{11} + \frac{17863246247253068318981803095547841904398727652618928114563402216368301467782200766669}{7060077644640734155449234049319619105512081512353313089689998258043091262778096584916992} a^{10} - \frac{64995823782996181313922278461869945533245135855174283939886296222005628394341616164151}{14120155289281468310898468098639238211024163024706626179379996516086182525556193169833984} a^{9} - \frac{3119497004717931602428615145586711485652156871745936119879097201347263919526609232489}{1765019411160183538862308512329904776378020378088328272422499564510772815694524146229248} a^{8} - \frac{32824251331442176460076538352570741949896547617417420456671108704969977684395324921073}{3530038822320367077724617024659809552756040756176656544844999129021545631389048292458496} a^{7} + \frac{35859225115870558814197368999289969979312314247245661057405827470894068751836552153893}{882509705580091769431154256164952388189010189044164136211249782255386407847262073114624} a^{6} - \frac{10127725534642210061371873587140681050729229604913190569896218542033808613494744456261}{441254852790045884715577128082476194094505094522082068105624891127693203923631036557312} a^{5} - \frac{16905535829263276017344748592577239847611553377020083667002778782243981041948964724197}{220627426395022942357788564041238097047252547261041034052812445563846601961815518278656} a^{4} - \frac{52749253524910765235510679981073100583859468442534279436688564833039161054088794378325}{220627426395022942357788564041238097047252547261041034052812445563846601961815518278656} a^{3} + \frac{3639541477039902899492594524803770856403124181540919911073533379356152559534030442791}{55156856598755735589447141010309524261813136815260258513203111390961650490453879569664} a^{2} - \frac{20749863727923427324257482400724481184804835231402523899928691058018747754826203924021}{55156856598755735589447141010309524261813136815260258513203111390961650490453879569664} a - \frac{2213620763679154123681629745782954415627606207085663372500140660655498818000916725359}{13789214149688933897361785252577381065453284203815064628300777847740412622613469892416}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, 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:  \( 1295218331170000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.280732967047165372057.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$71$71.4.3.2$x^{4} - 71$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.2$x^{4} - 71$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.2$x^{4} - 71$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.2$x^{4} - 71$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$