↑

Properties

Label 16.16.1535944563...5041.1
Degree $16$
Signature $[16, 0]$
Discriminant $41^{15}\cdot 61^{14}$
Root discriminant $1186.17$
Ramified primes $41, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3477032665088, 7374113275904, -10929645040640, -14649824518528, 14316222629824, 475008820192, -1122845915066, -27058405023, 27172084547, 537783345, -260470219, -2578770, 1025838, 5733, -1709, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 1709*x^14 + 5733*x^13 + 1025838*x^12 - 2578770*x^11 - 260470219*x^10 + 537783345*x^9 + 27172084547*x^8 - 27058405023*x^7 - 1122845915066*x^6 + 475008820192*x^5 + 14316222629824*x^4 - 14649824518528*x^3 - 10929645040640*x^2 + 7374113275904*x + 3477032665088)
 
gp: K = bnfinit(x^16 - 5*x^15 - 1709*x^14 + 5733*x^13 + 1025838*x^12 - 2578770*x^11 - 260470219*x^10 + 537783345*x^9 + 27172084547*x^8 - 27058405023*x^7 - 1122845915066*x^6 + 475008820192*x^5 + 14316222629824*x^4 - 14649824518528*x^3 - 10929645040640*x^2 + 7374113275904*x + 3477032665088, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 1709 x^{14} + 5733 x^{13} + 1025838 x^{12} - 2578770 x^{11} - 260470219 x^{10} + 537783345 x^{9} + 27172084547 x^{8} - 27058405023 x^{7} - 1122845915066 x^{6} + 475008820192 x^{5} + 14316222629824 x^{4} - 14649824518528 x^{3} - 10929645040640 x^{2} + 7374113275904 x + 3477032665088 \)

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:  \(15359445630083788418852698524374845584773320185041=41^{15}\cdot 61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1186.17$
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:  $41, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4} - \frac{5}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} + \frac{1}{16} a^{7} - \frac{1}{4} a^{6} + \frac{5}{32} a^{5} - \frac{3}{16} a^{4} - \frac{15}{32} a^{3} + \frac{7}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{12} + \frac{1}{64} a^{10} - \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} + \frac{9}{64} a^{6} - \frac{5}{32} a^{5} + \frac{15}{64} a^{4} - \frac{7}{16} a^{3} - \frac{7}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} - \frac{1}{128} a^{11} + \frac{1}{128} a^{10} + \frac{1}{64} a^{9} + \frac{3}{64} a^{8} + \frac{13}{128} a^{7} + \frac{5}{128} a^{6} - \frac{25}{128} a^{5} - \frac{3}{128} a^{4} + \frac{5}{64} a^{3} - \frac{1}{16} a^{2}$, $\frac{1}{1024} a^{14} + \frac{3}{1024} a^{13} - \frac{5}{1024} a^{12} - \frac{3}{1024} a^{11} - \frac{13}{512} a^{10} + \frac{31}{512} a^{9} - \frac{59}{1024} a^{8} - \frac{167}{1024} a^{7} - \frac{101}{1024} a^{6} - \frac{39}{1024} a^{5} - \frac{33}{512} a^{4} + \frac{17}{64} a^{3} + \frac{1}{4} a^{2} - \frac{3}{8} a$, $\frac{1}{2025258846665401579448276770613269918779379572160272030264822472876862005648393951684395008} a^{15} - \frac{892598287458002703739397241053279209082036300036713042300620721944009559199889314637141}{2025258846665401579448276770613269918779379572160272030264822472876862005648393951684395008} a^{14} + \frac{7722547352305913865087858850921349169474293740538626331685813738564486366232316966332131}{2025258846665401579448276770613269918779379572160272030264822472876862005648393951684395008} a^{13} - \frac{12605594526149917267860534617038300425797824881395918890361006599890396148889716883024267}{2025258846665401579448276770613269918779379572160272030264822472876862005648393951684395008} a^{12} - \frac{980268067420429637287951909246113259978750264197356508984385100428483116462842380867633}{1012629423332700789724138385306634959389689786080136015132411236438431002824196975842197504} a^{11} + \frac{6236800555343989126438594380761001646131788610726010770194342677307891864582908835019943}{1012629423332700789724138385306634959389689786080136015132411236438431002824196975842197504} a^{10} + \frac{53374879052735108190622773505409225301834281414851271614141376670336721062985456126984149}{2025258846665401579448276770613269918779379572160272030264822472876862005648393951684395008} a^{9} - \frac{12835969995672785365845084824727961989104609117711744894654924457988815199636260063818591}{2025258846665401579448276770613269918779379572160272030264822472876862005648393951684395008} a^{8} + \frac{433516497206759093330679522269514583180683061602640585990155500411134535416729315650041331}{2025258846665401579448276770613269918779379572160272030264822472876862005648393951684395008} a^{7} + \frac{408679491609214406672285232955818932614649837160123073818419015911202186404567211771484273}{2025258846665401579448276770613269918779379572160272030264822472876862005648393951684395008} a^{6} + \frac{68911787881394420936294161859933390865885506008218395236142722873492737094288766773545723}{1012629423332700789724138385306634959389689786080136015132411236438431002824196975842197504} a^{5} - \frac{1875406231145441892223942033337403691829234627650643523104981797746944740027431250549221}{7911167369786724919719831135208085620231951453751062618221962784675242209564038873767168} a^{4} + \frac{2648667803924127837942248034161656436315879076466667040485175539113519813696454825343791}{31644669479146899678879324540832342480927805815004250472887851138700968838256155495068672} a^{3} - \frac{6573790556185189386081873744892516747334775313243287778879287681922485114092439805852091}{15822334739573449839439662270416171240463902907502125236443925569350484419128077747534336} a^{2} - \frac{182999387353656890288051178346952403566893553572272088653814924717711614746239171909039}{1977791842446681229929957783802021405057987863437765654555490696168810552391009718441792} a + \frac{42512940550685566924723845879910273480426067404063729252564530309369176522656651782391}{123611990152917576870622361487626337816124241464860353409718168510550659524438107402612}$

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:  $15$
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:  \( 4032996399580000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.256455041.2, 8.8.10033813098753844365041.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
41Data not computed
$61$61.8.7.4$x^{8} + 488$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.4$x^{8} + 488$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$