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Properties

Label 16.8.20027199604...9569.1
Degree $16$
Signature $[8, 4]$
Discriminant $71^{12}\cdot 73^{14}$
Root discriminant $1044.36$
Ramified primes $71, 73$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (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![3128974954972, 2869903672164, -2815961493701, -114161384246, 72782355414, -18864989696, 6319219283, -11556770, -15878925, 12694968, -2719805, 95962, -13749, -1064, 172, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 - 13749*x^12 + 95962*x^11 - 2719805*x^10 + 12694968*x^9 - 15878925*x^8 - 11556770*x^7 + 6319219283*x^6 - 18864989696*x^5 + 72782355414*x^4 - 114161384246*x^3 - 2815961493701*x^2 + 2869903672164*x + 3128974954972)
 
gp: K = bnfinit(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 - 13749*x^12 + 95962*x^11 - 2719805*x^10 + 12694968*x^9 - 15878925*x^8 - 11556770*x^7 + 6319219283*x^6 - 18864989696*x^5 + 72782355414*x^4 - 114161384246*x^3 - 2815961493701*x^2 + 2869903672164*x + 3128974954972, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 172 x^{14} - 1064 x^{13} - 13749 x^{12} + 95962 x^{11} - 2719805 x^{10} + 12694968 x^{9} - 15878925 x^{8} - 11556770 x^{7} + 6319219283 x^{6} - 18864989696 x^{5} + 72782355414 x^{4} - 114161384246 x^{3} - 2815961493701 x^{2} + 2869903672164 x + 3128974954972 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{142} a^{4} - \frac{1}{71} a^{3} - \frac{17}{71} a^{2} + \frac{35}{142} a + \frac{20}{71}$, $\frac{1}{142} a^{5} - \frac{19}{71} a^{3} - \frac{33}{142} a^{2} - \frac{16}{71} a - \frac{31}{71}$, $\frac{1}{568} a^{6} + \frac{1}{568} a^{5} + \frac{1}{568} a^{4} - \frac{149}{568} a^{3} - \frac{255}{568} a^{2} - \frac{7}{568} a - \frac{103}{284}$, $\frac{1}{568} a^{7} - \frac{1}{284} a^{4} + \frac{83}{284} a^{3} - \frac{30}{71} a^{2} - \frac{131}{568} a - \frac{61}{284}$, $\frac{1}{362952} a^{8} - \frac{1}{90738} a^{7} + \frac{13}{60492} a^{6} - \frac{55}{90738} a^{5} - \frac{223}{90738} a^{4} + \frac{1073}{181476} a^{3} + \frac{93787}{362952} a^{2} - \frac{1318}{5041} a + \frac{755}{90738}$, $\frac{1}{362952} a^{9} + \frac{31}{181476} a^{7} + \frac{23}{90738} a^{6} + \frac{98}{45369} a^{5} + \frac{63}{20164} a^{4} + \frac{131}{362952} a^{3} + \frac{13625}{45369} a^{2} - \frac{743}{45369} a - \frac{5519}{45369}$, $\frac{1}{725904} a^{10} - \frac{1}{725904} a^{9} - \frac{1}{725904} a^{8} + \frac{47}{120984} a^{7} - \frac{97}{181476} a^{6} + \frac{19}{90738} a^{5} - \frac{2317}{725904} a^{4} + \frac{204989}{725904} a^{3} + \frac{30361}{725904} a^{2} - \frac{2753}{120984} a - \frac{6416}{45369}$, $\frac{1}{725904} a^{11} - \frac{1}{725904} a^{8} - \frac{11}{60492} a^{7} - \frac{161}{362952} a^{6} + \frac{153}{80656} a^{5} + \frac{119}{362952} a^{4} + \frac{125135}{362952} a^{3} - \frac{8005}{80656} a^{2} + \frac{51721}{362952} a + \frac{5447}{45369}$, $\frac{1}{3999852991872} a^{12} - \frac{1}{666642165312} a^{11} + \frac{41}{1333284330624} a^{10} - \frac{35}{249990811992} a^{9} - \frac{3472903}{3999852991872} a^{8} + \frac{6947453}{1999926495936} a^{7} - \frac{271204583}{3999852991872} a^{6} + \frac{191244823}{999963247968} a^{5} + \frac{11610765179}{3999852991872} a^{4} - \frac{4080047339}{666642165312} a^{3} + \frac{206593728287}{1333284330624} a^{2} - \frac{16872662841}{111107027552} a + \frac{228553569869}{999963247968}$, $\frac{1}{147994560699264} a^{13} + \frac{1}{12332880058272} a^{12} - \frac{33061003}{49331520233088} a^{11} - \frac{33060181}{73997280349632} a^{10} + \frac{161822057}{147994560699264} a^{9} - \frac{7724891}{12332880058272} a^{8} + \frac{57747505037}{147994560699264} a^{7} + \frac{26655133847}{73997280349632} a^{6} + \frac{87711040433}{49331520233088} a^{5} + \frac{119890236229}{36998640174816} a^{4} - \frac{63907754478631}{147994560699264} a^{3} + \frac{6067219377307}{73997280349632} a^{2} + \frac{12890891885783}{36998640174816} a + \frac{2052367363121}{18499320087408}$, $\frac{1}{295989121398528} a^{14} - \frac{1}{295989121398528} a^{13} + \frac{17}{147994560699264} a^{12} - \frac{187345303}{295989121398528} a^{11} + \frac{10802651}{18499320087408} a^{10} - \frac{245883961}{295989121398528} a^{9} + \frac{45855289}{36998640174816} a^{8} - \frac{115559483365}{295989121398528} a^{7} - \frac{6832047173}{24665760116544} a^{6} + \frac{458798724449}{295989121398528} a^{5} - \frac{37679814347}{16443840077696} a^{4} + \frac{97811126392355}{295989121398528} a^{3} + \frac{54856729234433}{295989121398528} a^{2} + \frac{1650210130279}{6166440029136} a + \frac{1255279813753}{8221920038848}$, $\frac{1}{797098703926235904} a^{15} + \frac{1339}{797098703926235904} a^{14} + \frac{547}{199274675981558976} a^{13} + \frac{24247}{265699567975411968} a^{12} + \frac{35087354047}{398549351963117952} a^{11} + \frac{126992983301}{265699567975411968} a^{10} - \frac{159348707629}{132849783987705984} a^{9} - \frac{19401134985}{88566522658470656} a^{8} - \frac{109897338960289}{398549351963117952} a^{7} - \frac{582861789061663}{797098703926235904} a^{6} - \frac{211986286119697}{99637337990779488} a^{5} + \frac{2348252964283063}{797098703926235904} a^{4} + \frac{37569703816735585}{265699567975411968} a^{3} + \frac{5293693868490379}{22141630664617664} a^{2} + \frac{54804452261816863}{199274675981558976} a + \frac{3460440090577457}{49818668995389744}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 8928632422870000000 \) (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$