↑

Properties

Label 16.0.41061437092...6689.5
Degree $16$
Signature $[0, 8]$
Discriminant $29^{14}\cdot 53^{14}$
Root discriminant $614.24$
Ramified primes $29, 53$
Class number $38887696$ (GRH)
Class group $[2, 3118, 6236]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![231877632771072, -169299154243200, 49990424556304, -8627965710464, 1304389721708, -188022386120, 20717625413, -1902825239, 76707714, 11221001, -283221, 46132, 2855, -1265, 146, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 146*x^14 - 1265*x^13 + 2855*x^12 + 46132*x^11 - 283221*x^10 + 11221001*x^9 + 76707714*x^8 - 1902825239*x^7 + 20717625413*x^6 - 188022386120*x^5 + 1304389721708*x^4 - 8627965710464*x^3 + 49990424556304*x^2 - 169299154243200*x + 231877632771072)
 
gp: K = bnfinit(x^16 - 5*x^15 + 146*x^14 - 1265*x^13 + 2855*x^12 + 46132*x^11 - 283221*x^10 + 11221001*x^9 + 76707714*x^8 - 1902825239*x^7 + 20717625413*x^6 - 188022386120*x^5 + 1304389721708*x^4 - 8627965710464*x^3 + 49990424556304*x^2 - 169299154243200*x + 231877632771072, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 146 x^{14} - 1265 x^{13} + 2855 x^{12} + 46132 x^{11} - 283221 x^{10} + 11221001 x^{9} + 76707714 x^{8} - 1902825239 x^{7} + 20717625413 x^{6} - 188022386120 x^{5} + 1304389721708 x^{4} - 8627965710464 x^{3} + 49990424556304 x^{2} - 169299154243200 x + 231877632771072 \)

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:  \(410614370925299326221754150224933859206386689=29^{14}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $614.24$
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:  $29, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{12} a^{2} - \frac{1}{3} a$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{5}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{72} a^{11} - \frac{1}{72} a^{10} + \frac{1}{36} a^{9} - \frac{1}{36} a^{8} + \frac{1}{36} a^{7} + \frac{5}{36} a^{6} + \frac{11}{72} a^{5} + \frac{7}{72} a^{4} - \frac{5}{18} a^{3} + \frac{7}{36} a^{2} + \frac{1}{3} a$, $\frac{1}{144} a^{12} - \frac{1}{72} a^{10} - \frac{1}{48} a^{9} - \frac{1}{24} a^{8} - \frac{1}{4} a^{7} - \frac{7}{48} a^{6} + \frac{1}{6} a^{5} + \frac{13}{72} a^{4} + \frac{7}{16} a^{3} - \frac{5}{72} a^{2} - \frac{1}{12} a$, $\frac{1}{864} a^{13} - \frac{1}{864} a^{12} + \frac{1}{288} a^{10} - \frac{5}{864} a^{9} - \frac{17}{432} a^{8} - \frac{125}{864} a^{7} + \frac{101}{864} a^{6} - \frac{1}{72} a^{5} - \frac{49}{288} a^{4} - \frac{383}{864} a^{3} + \frac{199}{432} a^{2} - \frac{23}{72} a - \frac{1}{3}$, $\frac{1}{1728} a^{14} - \frac{1}{1728} a^{13} - \frac{1}{288} a^{12} + \frac{1}{576} a^{11} - \frac{29}{1728} a^{10} - \frac{13}{432} a^{9} - \frac{17}{1728} a^{8} + \frac{29}{1728} a^{7} + \frac{31}{288} a^{6} + \frac{23}{576} a^{5} + \frac{73}{1728} a^{4} + \frac{17}{108} a^{3} + \frac{1}{8} a^{2} - \frac{11}{24} a$, $\frac{1}{4138751314600143054660169726420512317001938777413042382939826768173312693026976904576} a^{15} + \frac{646628447706980249347495401012336008700668981895303381455963012165531248097961911}{4138751314600143054660169726420512317001938777413042382939826768173312693026976904576} a^{14} + \frac{1114361277875847751299439478916016365512492904086681210620692893227833643419612331}{2069375657300071527330084863210256158500969388706521191469913384086656346513488452288} a^{13} + \frac{5524481791404630607417312527381392341248460102148711042469576385850412799851930919}{4138751314600143054660169726420512317001938777413042382939826768173312693026976904576} a^{12} - \frac{26342293764191157763494299000259598891328932921934327608509752615172395884039570949}{4138751314600143054660169726420512317001938777413042382939826768173312693026976904576} a^{11} + \frac{4048829663661869468652147079540831944714544282693559451404210225362882843939931443}{517343914325017881832521215802564039625242347176630297867478346021664086628372113072} a^{10} - \frac{3066668615677632465042171651954705827100193347826222724588432529335873087007547687}{1379583771533381018220056575473504105667312925804347460979942256057770897675658968192} a^{9} - \frac{3611841925740172261465109387724643916539308573817310152217704328531746520972276009}{153287085725931224246672952830389345074145880644927495664438028450863433075073218688} a^{8} - \frac{80494879273461122447872692292694579290661807180969689550481824474265204746989962713}{2069375657300071527330084863210256158500969388706521191469913384086656346513488452288} a^{7} - \frac{923492821143203030814183065822551451115453237459491063155997810485968211690277460767}{4138751314600143054660169726420512317001938777413042382939826768173312693026976904576} a^{6} - \frac{260556290249616874514173595438714433564150760417370185600985020559998710710983946639}{4138751314600143054660169726420512317001938777413042382939826768173312693026976904576} a^{5} - \frac{192661840804008274295735246634458977477694860875041140309151474415775902929607992503}{1034687828650035763665042431605128079250484694353260595734956692043328173256744226144} a^{4} + \frac{138973372923624969222537442013074027390699151894234125291818834411891254369332268303}{1034687828650035763665042431605128079250484694353260595734956692043328173256744226144} a^{3} - \frac{17987586169156090560290694719938195678058960846176317978369322009664308939067017465}{129335978581254470458130303950641009906310586794157574466869586505416021657093028268} a^{2} + \frac{5487283318356776967562623575464254456141995964077877460555912391911757707424687525}{86223985720836313638753535967094006604207057862771716311246391003610681104728685512} a + \frac{220308735965208160058121960079737649714656269249231266765921225822319894516557081}{3592666071701513068281397331962250275175294077615488179635266291817111712697028563}$

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

Class group and class number

$C_{2}\times C_{3118}\times C_{6236}$, which has order $38887696$ (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:  \( 1358079502.6 \) (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

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{1537}) \), \(\Q(\sqrt{29}) \), 4.4.3630961153.1, 4.4.3630961153.2, \(\Q(\sqrt{29}, \sqrt{53})\), 8.0.20263621860992652421633.1 x2, 8.0.20263621860992652421633.2 x2, 8.8.13183878894595089409.1

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

Sibling fields

Galois closure: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$29$29.8.7.1$x^{8} - 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.1$x^{8} - 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$