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Properties

Label 16.0.12248930678...4361.7
Degree $16$
Signature $[0, 8]$
Discriminant $31^{12}\cdot 41^{15}$
Root discriminant $427.08$
Ramified primes $31, 41$
Class number $33280$ (GRH)
Class group $[8, 4160]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![227541079872, 47330809200, 2592547372, -35655384702, -7327431717, 2042810745, 1385098477, 250742109, 21634972, -622524, -576572, -125221, -4637, 1407, 48, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 48*x^14 + 1407*x^13 - 4637*x^12 - 125221*x^11 - 576572*x^10 - 622524*x^9 + 21634972*x^8 + 250742109*x^7 + 1385098477*x^6 + 2042810745*x^5 - 7327431717*x^4 - 35655384702*x^3 + 2592547372*x^2 + 47330809200*x + 227541079872)
 
gp: K = bnfinit(x^16 - 2*x^15 + 48*x^14 + 1407*x^13 - 4637*x^12 - 125221*x^11 - 576572*x^10 - 622524*x^9 + 21634972*x^8 + 250742109*x^7 + 1385098477*x^6 + 2042810745*x^5 - 7327431717*x^4 - 35655384702*x^3 + 2592547372*x^2 + 47330809200*x + 227541079872, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 48 x^{14} + 1407 x^{13} - 4637 x^{12} - 125221 x^{11} - 576572 x^{10} - 622524 x^{9} + 21634972 x^{8} + 250742109 x^{7} + 1385098477 x^{6} + 2042810745 x^{5} - 7327431717 x^{4} - 35655384702 x^{3} + 2592547372 x^{2} + 47330809200 x + 227541079872 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1224893067851117776958601403290228616174361=31^{12}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $427.08$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{208} a^{13} - \frac{23}{208} a^{12} + \frac{1}{13} a^{11} - \frac{3}{26} a^{10} - \frac{25}{208} a^{9} + \frac{1}{26} a^{8} + \frac{3}{16} a^{7} - \frac{75}{208} a^{6} + \frac{11}{104} a^{5} + \frac{7}{52} a^{4} + \frac{75}{208} a^{3} + \frac{47}{104} a^{2} - \frac{11}{52} a + \frac{1}{13}$, $\frac{1}{208} a^{14} + \frac{7}{208} a^{12} - \frac{5}{52} a^{11} - \frac{5}{208} a^{10} + \frac{5}{208} a^{9} + \frac{15}{208} a^{8} + \frac{47}{104} a^{7} - \frac{7}{16} a^{6} - \frac{45}{104} a^{5} - \frac{61}{208} a^{4} - \frac{53}{208} a^{3} + \frac{45}{104} a^{2} + \frac{11}{52} a - \frac{3}{13}$, $\frac{1}{4403897020962801250807697039392279183417242259647356896118590877783518548197808} a^{15} - \frac{958501273419685530071710220278155516925662360995221043712926468320201492823}{4403897020962801250807697039392279183417242259647356896118590877783518548197808} a^{14} - \frac{184394323748077448209465154885924402193419834410771536502279662926419866339}{733982836827133541801282839898713197236207043274559482686431812963919758032968} a^{13} - \frac{27478790130675917718851011914223707252819378039464441026192435774067605172587}{733982836827133541801282839898713197236207043274559482686431812963919758032968} a^{12} - \frac{224504746647061310072631367678213700780598139410662136728526133521381216341969}{4403897020962801250807697039392279183417242259647356896118590877783518548197808} a^{11} - \frac{58140476686015526094947882712885070849358788033600521825676210396325921219453}{550487127620350156350962129924034897927155282455919612014823859722939818524726} a^{10} - \frac{340591645470667406912958982264364648105311637729175099944309665405510750028351}{4403897020962801250807697039392279183417242259647356896118590877783518548197808} a^{9} - \frac{353369248081386397162724721115607461084569886766320648174791152252390385923897}{1467965673654267083602565679797426394472414086549118965372863625927839516065936} a^{8} + \frac{73131099097552689256844744235308875226861874895405434352982953351158656835413}{550487127620350156350962129924034897927155282455919612014823859722939818524726} a^{7} + \frac{456532838798060872481361764020224785849161153747465811457367967128590037687}{3321189306910106523987705157912729399258855399432395849259872456850315647208} a^{6} - \frac{58709096240406507676525679887170258052138989170505678158460620363319206426521}{259052765938988308871041002317192893142190721155726876242270051634324620482224} a^{5} + \frac{1039205976817421098597682256454689587803583894681875450092615986021072729153}{4343093709036293146753152898808953829800041676180825341339833212804258923272} a^{4} - \frac{189466912728605812167400773576737517370168420443478686729081848469158135893235}{733982836827133541801282839898713197236207043274559482686431812963919758032968} a^{3} + \frac{38590861544433607919625408636907165138330446890012418950873253648844135417143}{366991418413566770900641419949356598618103521637279741343215906481959879016484} a^{2} - \frac{255810397734750753694133717420838888205781841846828327433454154486350676220957}{550487127620350156350962129924034897927155282455919612014823859722939818524726} a + \frac{26851837093982215573689829557656445071741029605718603587975941273987854811862}{91747854603391692725160354987339149654525880409319935335803976620489969754121}$

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

Class group and class number

$C_{8}\times C_{4160}$, which has order $33280$ (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:  $7$
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:  \( 166813427270 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1271}) \), 4.0.66233081.1, 8.0.179859661768855001.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$31$31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed