↑

Properties

Label 16.8.80341646997...4929.1
Degree $16$
Signature $[8, 4]$
Discriminant $37^{12}\cdot 73^{14}$
Root discriminant $640.56$
Ramified primes $37, 73$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-179786105721269, 82626919055977, 25781607633385, -14072956021738, -2055904962519, -154664551218, 49129870727, 15221652842, -16641328, -222937160, -19740121, 712517, 245657, 2223, -934, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 934*x^14 + 2223*x^13 + 245657*x^12 + 712517*x^11 - 19740121*x^10 - 222937160*x^9 - 16641328*x^8 + 15221652842*x^7 + 49129870727*x^6 - 154664551218*x^5 - 2055904962519*x^4 - 14072956021738*x^3 + 25781607633385*x^2 + 82626919055977*x - 179786105721269)
 
gp: K = bnfinit(x^16 - 5*x^15 - 934*x^14 + 2223*x^13 + 245657*x^12 + 712517*x^11 - 19740121*x^10 - 222937160*x^9 - 16641328*x^8 + 15221652842*x^7 + 49129870727*x^6 - 154664551218*x^5 - 2055904962519*x^4 - 14072956021738*x^3 + 25781607633385*x^2 + 82626919055977*x - 179786105721269, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 934 x^{14} + 2223 x^{13} + 245657 x^{12} + 712517 x^{11} - 19740121 x^{10} - 222937160 x^{9} - 16641328 x^{8} + 15221652842 x^{7} + 49129870727 x^{6} - 154664551218 x^{5} - 2055904962519 x^{4} - 14072956021738 x^{3} + 25781607633385 x^{2} + 82626919055977 x - 179786105721269 \)

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:  \(803416469975725073264940954221144185207724929=37^{12}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $640.56$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{1515233047980573122969499301144147610332089543974057334474786705772288429144105855057247273694154225272644} a^{15} + \frac{42193736106232467080342581976458772610691619296208333341291937726146556298019681258143393237190664112227}{378808261995143280742374825286036902583022385993514333618696676443072107286026463764311818423538556318161} a^{14} - \frac{55378613380311861169006297691743347923780078056278183602950847920880688120199883964900055559115923476819}{757616523990286561484749650572073805166044771987028667237393352886144214572052927528623636847077112636322} a^{13} - \frac{119419154068396565559630200865385414456543191135766622461295791148632983163524232684716427789066335260521}{1515233047980573122969499301144147610332089543974057334474786705772288429144105855057247273694154225272644} a^{12} - \frac{157828142004005428836247046188003668767541665933682299638593087162538610529377068710020934337361862727105}{1515233047980573122969499301144147610332089543974057334474786705772288429144105855057247273694154225272644} a^{11} + \frac{219100546596877621543942059983334660253586067837410031610682986378138211505541675049491554763729811051529}{1515233047980573122969499301144147610332089543974057334474786705772288429144105855057247273694154225272644} a^{10} + \frac{51394596185099619036795286884492481332936189502649904344261000916209087332791577625602652814970523008241}{757616523990286561484749650572073805166044771987028667237393352886144214572052927528623636847077112636322} a^{9} - \frac{88220960190193091276170827756169046167159545750153145504896117248677265390114242563430671035707216965000}{378808261995143280742374825286036902583022385993514333618696676443072107286026463764311818423538556318161} a^{8} + \frac{596132481133427669351428312266264224943361492799341533009288644442589949641882770671198261327339224086349}{1515233047980573122969499301144147610332089543974057334474786705772288429144105855057247273694154225272644} a^{7} + \frac{179334556802908974408474921342467320999033001736636549197854300088194194667714195434908850912314719862143}{1515233047980573122969499301144147610332089543974057334474786705772288429144105855057247273694154225272644} a^{6} + \frac{531738049893179781570521055184346272398690564997033770196768167958830696951581265405215674197410779895339}{1515233047980573122969499301144147610332089543974057334474786705772288429144105855057247273694154225272644} a^{5} + \frac{133674841872178555535879864477220428106171745156643839194085459437159197925631363847819076678381636479123}{378808261995143280742374825286036902583022385993514333618696676443072107286026463764311818423538556318161} a^{4} - \frac{120356961387667331042132415986617095502111616494408414321676172094317137206419637331225148936135048931419}{757616523990286561484749650572073805166044771987028667237393352886144214572052927528623636847077112636322} a^{3} - \frac{26989791073766238634003444414597459585203581164605443265122954671960045320493557476418292776565169627977}{1515233047980573122969499301144147610332089543974057334474786705772288429144105855057247273694154225272644} a^{2} - \frac{61339143182481808288754538652502113844096019361736022661309839959091098226098357847906630307013762457835}{1515233047980573122969499301144147610332089543974057334474786705772288429144105855057247273694154225272644} a - \frac{130167312370373455825199196687600201793066701469748123780324127798116596497248520648404113610250655237677}{378808261995143280742374825286036902583022385993514333618696676443072107286026463764311818423538556318161}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $11$
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:  \( 7995947287160000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

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 $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{73}) \), \(\Q(\sqrt{2701}) \), \(\Q(\sqrt{37}) \), 4.4.532564273.1, 4.4.389017.1, \(\Q(\sqrt{37}, \sqrt{73})\), 8.8.283624704876018529.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
$73$73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$