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Properties

Label 16.16.7882353535...4481.1
Degree $16$
Signature $[16, 0]$
Discriminant $37^{12}\cdot 149^{12}$
Root discriminant $639.80$
Ramified primes $37, 149$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $Q_{16}$ (as 16T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-134701390338101, -1532647000809550, 1450005309994863, -408109365395354, 713563290839, 17178191896767, -1854337101495, -194082565895, 33537176305, 944435243, -251203331, -2632325, 913132, 5036, -1567, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 1567*x^14 + 5036*x^13 + 913132*x^12 - 2632325*x^11 - 251203331*x^10 + 944435243*x^9 + 33537176305*x^8 - 194082565895*x^7 - 1854337101495*x^6 + 17178191896767*x^5 + 713563290839*x^4 - 408109365395354*x^3 + 1450005309994863*x^2 - 1532647000809550*x - 134701390338101)
 
gp: K = bnfinit(x^16 - 5*x^15 - 1567*x^14 + 5036*x^13 + 913132*x^12 - 2632325*x^11 - 251203331*x^10 + 944435243*x^9 + 33537176305*x^8 - 194082565895*x^7 - 1854337101495*x^6 + 17178191896767*x^5 + 713563290839*x^4 - 408109365395354*x^3 + 1450005309994863*x^2 - 1532647000809550*x - 134701390338101, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 1567 x^{14} + 5036 x^{13} + 913132 x^{12} - 2632325 x^{11} - 251203331 x^{10} + 944435243 x^{9} + 33537176305 x^{8} - 194082565895 x^{7} - 1854337101495 x^{6} + 17178191896767 x^{5} + 713563290839 x^{4} - 408109365395354 x^{3} + 1450005309994863 x^{2} - 1532647000809550 x - 134701390338101 \)

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:  \(788235353584876852533663970039941030620064481=37^{12}\cdot 149^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $639.80$
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:  $37, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{266072926939294628691226261625935881971919389122856509385373603736903557895692914006243453149266424653} a^{15} - \frac{380588752706134433224783207564472278548511339016799198208963932321744618109698438279555335916781225}{3971237715511860129719794949640834059282378942132186707244382145326918774562580806063335121630842159} a^{14} - \frac{5134295669337470520648207559688260009925463169379460381259840673400271144396758748140953755328686026}{88690975646431542897075420541978627323973129707618836461791201245634519298564304668747817716422141551} a^{13} - \frac{6120426620488275652629320748055654155632050788798431083595561467961016283702819571910824785217652715}{266072926939294628691226261625935881971919389122856509385373603736903557895692914006243453149266424653} a^{12} - \frac{90112310805519817352267146844796964311607774665115137014180668596995954945007723509333220447625327001}{266072926939294628691226261625935881971919389122856509385373603736903557895692914006243453149266424653} a^{11} - \frac{956558789312586844822093388109769744027278636280203520136805018237310938159926010711305261429759000}{5020243904514992994174080408036526074941875266468990743120256674281199205579111585023461380174838201} a^{10} + \frac{14890935327313821804102260369673479292252893440948954565165689002036842388174991632715491756214087930}{266072926939294628691226261625935881971919389122856509385373603736903557895692914006243453149266424653} a^{9} - \frac{5688011842210369964787186689898308399980911789472526304080819776099050109434381640059646376982036171}{88690975646431542897075420541978627323973129707618836461791201245634519298564304668747817716422141551} a^{8} - \frac{14162268421084731083113152128126789465627299146318739287469785534371436730246821938930811238138553}{266072926939294628691226261625935881971919389122856509385373603736903557895692914006243453149266424653} a^{7} + \frac{128418474301599816775327222496531655772581704036592198208988751788103459755261931280592491080022626548}{266072926939294628691226261625935881971919389122856509385373603736903557895692914006243453149266424653} a^{6} - \frac{28613069799474700500099672881113900945895827685560776827279642220365611940380098216413932168468024125}{88690975646431542897075420541978627323973129707618836461791201245634519298564304668747817716422141551} a^{5} - \frac{13862798507636662090319904232995023749014994645250197481770404030342989310363842529462423957207329111}{266072926939294628691226261625935881971919389122856509385373603736903557895692914006243453149266424653} a^{4} + \frac{15313664772225452430972765459579685990010345913565244081218411297055005639729218296657122927600358835}{88690975646431542897075420541978627323973129707618836461791201245634519298564304668747817716422141551} a^{3} + \frac{77843543245770070747887409654178057918461955411687767094830519812524087051608781653294317021471120293}{266072926939294628691226261625935881971919389122856509385373603736903557895692914006243453149266424653} a^{2} + \frac{21331581445343389059333356994989214907528452272831227749595418805126083441248900092805307221983348275}{88690975646431542897075420541978627323973129707618836461791201245634519298564304668747817716422141551} a - \frac{12975106231825056143813526984178377207280805346233174332251916519121244362422884059688794163979153286}{266072926939294628691226261625935881971919389122856509385373603736903557895692914006243453149266424653}$

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:  \( 51119672626600000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_{16}$ (as 16T14):

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 $Q_{16}$
Character table for $Q_{16}$

Intermediate fields

\(\Q(\sqrt{5513}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{37}, \sqrt{149})\), 4.4.821437.1 x2, 4.4.203981.1 x2, 8.8.923744721862561.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
$149$149.4.3.2$x^{4} - 596$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.2$x^{4} - 596$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.2$x^{4} - 596$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.2$x^{4} - 596$$4$$1$$3$$C_4$$[\ ]_{4}$