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Properties

Label 16.16.2176717485...0161.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{14}\cdot 157^{14}$
Root discriminant $787.25$
Ramified primes $13, 157$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12486085888, 60542675904, 7339560560, -48328678280, -4542410360, 12242227394, 432751103, -1329168051, 35529210, 66746609, -5307893, -1327426, 163389, 3919, -826, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 826*x^14 + 3919*x^13 + 163389*x^12 - 1327426*x^11 - 5307893*x^10 + 66746609*x^9 + 35529210*x^8 - 1329168051*x^7 + 432751103*x^6 + 12242227394*x^5 - 4542410360*x^4 - 48328678280*x^3 + 7339560560*x^2 + 60542675904*x + 12486085888)
 
gp: K = bnfinit(x^16 - 5*x^15 - 826*x^14 + 3919*x^13 + 163389*x^12 - 1327426*x^11 - 5307893*x^10 + 66746609*x^9 + 35529210*x^8 - 1329168051*x^7 + 432751103*x^6 + 12242227394*x^5 - 4542410360*x^4 - 48328678280*x^3 + 7339560560*x^2 + 60542675904*x + 12486085888, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 826 x^{14} + 3919 x^{13} + 163389 x^{12} - 1327426 x^{11} - 5307893 x^{10} + 66746609 x^{9} + 35529210 x^{8} - 1329168051 x^{7} + 432751103 x^{6} + 12242227394 x^{5} - 4542410360 x^{4} - 48328678280 x^{3} + 7339560560 x^{2} + 60542675904 x + 12486085888 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21767174858780577201904716993499183222663790161=13^{14}\cdot 157^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $787.25$
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:  $13, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{56} a^{12} + \frac{1}{28} a^{11} - \frac{1}{14} a^{10} - \frac{1}{56} a^{9} - \frac{1}{28} a^{8} - \frac{1}{14} a^{7} - \frac{13}{56} a^{6} - \frac{1}{4} a^{5} + \frac{1}{14} a^{4} - \frac{3}{56} a^{3} - \frac{1}{28} a^{2} - \frac{1}{2} a + \frac{3}{7}$, $\frac{1}{1120} a^{13} + \frac{1}{560} a^{12} + \frac{1}{112} a^{11} - \frac{71}{1120} a^{10} - \frac{1}{70} a^{9} - \frac{3}{56} a^{8} - \frac{41}{1120} a^{7} + \frac{19}{80} a^{6} - \frac{15}{112} a^{5} - \frac{37}{224} a^{4} - \frac{39}{280} a^{3} - \frac{3}{20} a^{2} - \frac{11}{140} a - \frac{2}{5}$, $\frac{1}{15680} a^{14} - \frac{1}{7840} a^{13} - \frac{19}{7840} a^{12} + \frac{649}{15680} a^{11} - \frac{313}{3920} a^{10} - \frac{129}{3920} a^{9} + \frac{1399}{15680} a^{8} + \frac{59}{1568} a^{7} - \frac{1187}{7840} a^{6} + \frac{475}{3136} a^{5} + \frac{123}{1960} a^{4} - \frac{61}{245} a^{3} + \frac{83}{1960} a^{2} + \frac{207}{490} a - \frac{22}{245}$, $\frac{1}{66962209866020606016988158331697307198178239004987483669578560} a^{15} - \frac{55805784734450652364854679545635966755958196663636605195}{1913205996172017314771090238048494491376521114428213819130816} a^{14} - \frac{170144344908910504764972472803164791755047428483970031767}{4783014990430043286927725595121236228441302786070534547827040} a^{13} + \frac{577137749155018826180000590691288706517174147861192817983671}{66962209866020606016988158331697307198178239004987483669578560} a^{12} - \frac{3309617681867348817325399278832592730089124819090910607009719}{66962209866020606016988158331697307198178239004987483669578560} a^{11} - \frac{651852319753316902647199343845084105327663764304709676306117}{33481104933010303008494079165848653599089119502493741834789280} a^{10} - \frac{5663343598216195976351836302028313067427941039945536782215597}{66962209866020606016988158331697307198178239004987483669578560} a^{9} + \frac{51976941833774115601962846897193170737466311833424038049517}{1366575711551440939122207312891781779554657938877295585093440} a^{8} + \frac{3969283720712964576334222880401303489305652103216264502144921}{33481104933010303008494079165848653599089119502493741834789280} a^{7} + \frac{205105155781723314401454583202531115722360522550878919636803}{9566029980860086573855451190242472456882605572141069095654080} a^{6} - \frac{4940598506784867031861073318233794040940634269501752855730381}{66962209866020606016988158331697307198178239004987483669578560} a^{5} - \frac{5356789600621064495117872920999259998526773835011471072854083}{33481104933010303008494079165848653599089119502493741834789280} a^{4} + \frac{806594596790198509271065386534493413753031414381704224263277}{4185138116626287876061759895731081699886139937811717729348660} a^{3} + \frac{213631934178597687321138150625619815226106743765112850491}{239150749521502164346386279756061811422065139303526727391352} a^{2} + \frac{1930571529572110427810876154214245869555605772222917762371853}{4185138116626287876061759895731081699886139937811717729348660} a + \frac{176610338315453300819217733363139789218265348649097433933598}{1046284529156571969015439973932770424971534984452929432337165}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $15$
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:  \( 2472425742860000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{2041}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{157}) \), 4.4.8502154921.2, 4.4.8502154921.1, \(\Q(\sqrt{13}, \sqrt{157})\), 8.8.147537028771697097647881.2 x2, 8.8.147537028771697097647881.1 x2, 8.8.72286638300684516241.3

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$157$157.8.7.2$x^{8} - 3925$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
157.8.7.2$x^{8} - 3925$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$