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Properties

Label 16.8.72902183182...2777.1
Degree $16$
Signature $[8, 4]$
Discriminant $67^{12}\cdot 73^{15}$
Root discriminant $1307.44$
Ramified primes $67, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![513540167205851, 1159487494913896, -1164602131361688, 9508212706382, -2945197729257, -2188176847700, 758427128184, -24667276606, 5889687494, 188587700, -41550754, 2090766, -348370, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 348370*x^12 + 2090766*x^11 - 41550754*x^10 + 188587700*x^9 + 5889687494*x^8 - 24667276606*x^7 + 758427128184*x^6 - 2188176847700*x^5 - 2945197729257*x^4 + 9508212706382*x^3 - 1164602131361688*x^2 + 1159487494913896*x + 513540167205851)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 348370*x^12 + 2090766*x^11 - 41550754*x^10 + 188587700*x^9 + 5889687494*x^8 - 24667276606*x^7 + 758427128184*x^6 - 2188176847700*x^5 - 2945197729257*x^4 + 9508212706382*x^3 - 1164602131361688*x^2 + 1159487494913896*x + 513540167205851, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 348370 x^{12} + 2090766 x^{11} - 41550754 x^{10} + 188587700 x^{9} + 5889687494 x^{8} - 24667276606 x^{7} + 758427128184 x^{6} - 2188176847700 x^{5} - 2945197729257 x^{4} + 9508212706382 x^{3} - 1164602131361688 x^{2} + 1159487494913896 x + 513540167205851 \)

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:  \(72902183182742832065008845132818265359946436492777=67^{12}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1307.44$
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:  $67, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{134} a^{4} - \frac{1}{67} a^{3} - \frac{16}{67} a^{2} + \frac{33}{134} a + \frac{21}{134}$, $\frac{1}{134} a^{5} - \frac{18}{67} a^{3} - \frac{31}{134} a^{2} - \frac{47}{134} a + \frac{21}{67}$, $\frac{1}{134} a^{6} + \frac{31}{134} a^{3} + \frac{7}{134} a^{2} + \frac{12}{67} a - \frac{24}{67}$, $\frac{1}{134} a^{7} - \frac{65}{134} a^{3} - \frac{28}{67} a^{2} + \frac{1}{134} a + \frac{19}{134}$, $\frac{1}{35912} a^{8} - \frac{1}{8978} a^{7} + \frac{37}{17956} a^{6} + \frac{15}{8978} a^{5} - \frac{1}{8978} a^{4} - \frac{4649}{17956} a^{3} + \frac{16897}{35912} a^{2} - \frac{1049}{17956} a + \frac{4595}{35912}$, $\frac{1}{35912} a^{9} + \frac{29}{17956} a^{7} + \frac{11}{4489} a^{6} - \frac{4}{4489} a^{5} + \frac{33}{17956} a^{4} - \frac{1803}{35912} a^{3} - \frac{6383}{17956} a^{2} - \frac{11301}{35912} a + \frac{187}{4489}$, $\frac{1}{35912} a^{10} + \frac{13}{8978} a^{7} - \frac{9}{8978} a^{6} + \frac{35}{17956} a^{5} + \frac{37}{35912} a^{4} + \frac{4773}{17956} a^{3} + \frac{7509}{35912} a^{2} + \frac{1851}{8978} a - \frac{5017}{17956}$, $\frac{1}{35912} a^{11} - \frac{12}{4489} a^{7} - \frac{13}{17956} a^{6} + \frac{133}{35912} a^{5} + \frac{53}{17956} a^{4} - \frac{10423}{35912} a^{3} - \frac{1303}{4489} a^{2} + \frac{3301}{17956} a + \frac{1388}{4489}$, $\frac{1}{19248832} a^{12} - \frac{3}{9624416} a^{11} + \frac{23}{2406104} a^{10} + \frac{207}{19248832} a^{9} + \frac{77}{19248832} a^{8} + \frac{4187}{1203052} a^{7} - \frac{42853}{19248832} a^{6} + \frac{8693}{2406104} a^{5} + \frac{62051}{19248832} a^{4} + \frac{7782651}{19248832} a^{3} + \frac{6651365}{19248832} a^{2} + \frac{9584459}{19248832} a - \frac{8305037}{19248832}$, $\frac{1}{19248832} a^{13} + \frac{37}{4812208} a^{11} + \frac{239}{19248832} a^{10} + \frac{247}{19248832} a^{9} - \frac{41}{9624416} a^{8} - \frac{63269}{19248832} a^{7} - \frac{34827}{9624416} a^{6} - \frac{22381}{19248832} a^{5} - \frac{16899}{19248832} a^{4} + \frac{7331671}{19248832} a^{3} + \frac{3829737}{19248832} a^{2} + \frac{9620261}{19248832} a + \frac{466097}{9624416}$, $\frac{1}{174104987556391401790681816223847616} a^{14} - \frac{7}{174104987556391401790681816223847616} a^{13} - \frac{1324444896234886685599965277}{174104987556391401790681816223847616} a^{12} + \frac{7946669377409320113599791753}{174104987556391401790681816223847616} a^{11} + \frac{3638039485531176557017362239}{978117907620176414554392225976672} a^{10} + \frac{768700477712630984989161038711}{87052493778195700895340908111923808} a^{9} + \frac{434754991547581542612730567411}{43526246889097850447670454055961904} a^{8} + \frac{294185364043658009919209420094013}{174104987556391401790681816223847616} a^{7} + \frac{145835782364873542435033370897649}{87052493778195700895340908111923808} a^{6} + \frac{2972936351604094317181777792781}{2720390430568615652979403378497619} a^{5} - \frac{265445223845179988363032989588175}{174104987556391401790681816223847616} a^{4} - \frac{8124137614971011404989784058759379}{174104987556391401790681816223847616} a^{3} - \frac{27810783829583286342632220571356903}{174104987556391401790681816223847616} a^{2} + \frac{21245610823440923735747033059822537}{43526246889097850447670454055961904} a - \frac{62652353472733220561924245465373937}{174104987556391401790681816223847616}$, $\frac{1}{32932132501278990040109256220557000414016} a^{15} + \frac{11821}{4116516562659873755013657027569625051752} a^{14} - \frac{849260105270260490148121087762611}{32932132501278990040109256220557000414016} a^{13} - \frac{296546567046093269378334584226267}{16466066250639495020054628110278500207008} a^{12} + \frac{173210392924826383580785546278502049}{32932132501278990040109256220557000414016} a^{11} + \frac{440364761455267976378571333258058509}{32932132501278990040109256220557000414016} a^{10} - \frac{355937524725218293612701253427479601}{32932132501278990040109256220557000414016} a^{9} + \frac{267399815202376400082266252955721943}{32932132501278990040109256220557000414016} a^{8} - \frac{35237965799105414902453962704229581045}{16466066250639495020054628110278500207008} a^{7} + \frac{9760443971858801781765059182817165823}{4116516562659873755013657027569625051752} a^{6} + \frac{807124145718919573132698726594625085}{445028817584851216758233192169689194784} a^{5} - \frac{82008403754418421058560064007520356657}{32932132501278990040109256220557000414016} a^{4} - \frac{13915197929949483825201181881646841758251}{32932132501278990040109256220557000414016} a^{3} + \frac{4002699956651604811128914246561242587401}{16466066250639495020054628110278500207008} a^{2} - \frac{8175615994148245175827718013132896972053}{16466066250639495020054628110278500207008} a - \frac{2695138152483419264457499747181033343221}{32932132501278990040109256220557000414016}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 10058203138300000000 \) (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

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.222617464293544457737.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

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
$67$67.8.6.3$x^{8} - 67 x^{4} + 53868$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
67.8.6.3$x^{8} - 67 x^{4} + 53868$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
73Data not computed