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Properties

Label 16.0.26255811630...3841.3
Degree $16$
Signature $[0, 8]$
Discriminant $61^{14}\cdot 149^{14}$
Root discriminant $2908.72$
Ramified primes $61, 149$
Class number $8115887888$ (GRH)
Class group $[2, 2, 2028971972]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42985739425875246071, -14238289298038957907, 2883739495205635060, -268920195666810337, 32436664122096987, -1827680235740115, 171426115905879, -4376626157712, 356585283872, -3212165856, 818111467, 7456671, 657498, 28801, 842, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 842*x^14 + 28801*x^13 + 657498*x^12 + 7456671*x^11 + 818111467*x^10 - 3212165856*x^9 + 356585283872*x^8 - 4376626157712*x^7 + 171426115905879*x^6 - 1827680235740115*x^5 + 32436664122096987*x^4 - 268920195666810337*x^3 + 2883739495205635060*x^2 - 14238289298038957907*x + 42985739425875246071)
 
gp: K = bnfinit(x^16 - 5*x^15 + 842*x^14 + 28801*x^13 + 657498*x^12 + 7456671*x^11 + 818111467*x^10 - 3212165856*x^9 + 356585283872*x^8 - 4376626157712*x^7 + 171426115905879*x^6 - 1827680235740115*x^5 + 32436664122096987*x^4 - 268920195666810337*x^3 + 2883739495205635060*x^2 - 14238289298038957907*x + 42985739425875246071, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 842 x^{14} + 28801 x^{13} + 657498 x^{12} + 7456671 x^{11} + 818111467 x^{10} - 3212165856 x^{9} + 356585283872 x^{8} - 4376626157712 x^{7} + 171426115905879 x^{6} - 1827680235740115 x^{5} + 32436664122096987 x^{4} - 268920195666810337 x^{3} + 2883739495205635060 x^{2} - 14238289298038957907 x + 42985739425875246071 \)

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:  \(26255811630589850827051927649928110197766220114319683841=61^{14}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2908.72$
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:  $61, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{1275894581371252749824519641176879926458881309545791417107845640362503866247464831451875525728763944527136580013821610130625963100901091655} a^{15} + \frac{36553528639079406385407341259494386201435368781710670016988091373195901730355080064458418029929213286940360773883002634210172310405894062}{1275894581371252749824519641176879926458881309545791417107845640362503866247464831451875525728763944527136580013821610130625963100901091655} a^{14} + \frac{3670374841441375350577943796471997314964194684020226952561986791545858521461818523856894099301424625950445255151183584375723432431728172}{67152346387960671043395770588256838234677963660304811426728717913815992960392885865888185564671786554059820000727453164769787531626373245} a^{13} + \frac{75941282088507925531294549871333544327575495664573461603554738472057292458453693251821729214895334716240192764343873148913146069688917293}{1275894581371252749824519641176879926458881309545791417107845640362503866247464831451875525728763944527136580013821610130625963100901091655} a^{12} + \frac{597706276529415881164298881149212687604409564767366116522972512305206637050235936329163469537938606663141429819656075153825059427242901094}{1275894581371252749824519641176879926458881309545791417107845640362503866247464831451875525728763944527136580013821610130625963100901091655} a^{11} - \frac{535907708924310552529470724028346843188786130733657797646817332450271512353423741341358867366745131196102624925938214454185719899375620891}{1275894581371252749824519641176879926458881309545791417107845640362503866247464831451875525728763944527136580013821610130625963100901091655} a^{10} + \frac{123514407625757189688888681002345388242182614802912392267984660154807462083854215965940037903274600334134891971628017651534734889289442557}{255178916274250549964903928235375985291776261909158283421569128072500773249492966290375105145752788905427316002764322026125192620180218331} a^{9} + \frac{450498144340134194976687852803814709514635161986940933144461580118725278151567011811957876683100301268908151643678218249814072190584065266}{1275894581371252749824519641176879926458881309545791417107845640362503866247464831451875525728763944527136580013821610130625963100901091655} a^{8} + \frac{30720719976462506468359402663175284998315036389313717912302247461025438431215888674771366649479906655576037583749094122356098560108792492}{255178916274250549964903928235375985291776261909158283421569128072500773249492966290375105145752788905427316002764322026125192620180218331} a^{7} + \frac{33503260151445910757999415699742432823572039027821282183633477329774961411717779402036982373045115342160319339823245983362995783136977009}{67152346387960671043395770588256838234677963660304811426728717913815992960392885865888185564671786554059820000727453164769787531626373245} a^{6} + \frac{19590494693174789148907213659077089095856043858630992057825521882266678892971968438525149700503515871574986188965769385272583911449845553}{67152346387960671043395770588256838234677963660304811426728717913815992960392885865888185564671786554059820000727453164769787531626373245} a^{5} + \frac{10767577452602756771500664765278916604468383211244909930137868207685858488338924396799467332216668150591459562116644741642004571572614951}{255178916274250549964903928235375985291776261909158283421569128072500773249492966290375105145752788905427316002764322026125192620180218331} a^{4} + \frac{424019246421103338034048676627340886992287003512512737744904134054882298210017479640313391730059708622225669218586150589120526851423090987}{1275894581371252749824519641176879926458881309545791417107845640362503866247464831451875525728763944527136580013821610130625963100901091655} a^{3} + \frac{66249217630335987983109384264384209112444121410866145792660848106691010854188080462217164766265485292580203988827752081667119775605464877}{1275894581371252749824519641176879926458881309545791417107845640362503866247464831451875525728763944527136580013821610130625963100901091655} a^{2} - \frac{74850375863793630127158406024465958602618089880103982758293411390199158733005349358235263853914781952196686219471796718675522260018398603}{1275894581371252749824519641176879926458881309545791417107845640362503866247464831451875525728763944527136580013821610130625963100901091655} a + \frac{86934918195181861084552535035165449416487724258292876710794604007327119834733961185638527732858521946012835410972734924638287023897546206}{255178916274250549964903928235375985291776261909158283421569128072500773249492966290375105145752788905427316002764322026125192620180218331}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2028971972}$, which has order $8115887888$ (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:  \( 103227707426 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{9089}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{61}, \sqrt{149})\), 8.8.563763066196879006536961.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$149$149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$