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Properties

Label 16.0.12248930678...4361.5
Degree $16$
Signature $[0, 8]$
Discriminant $31^{12}\cdot 41^{15}$
Root discriminant $427.08$
Ramified primes $31, 41$
Class number $2560$ (GRH)
Class group $[8, 320]$ (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![190379307008, 444135049728, 494720800128, 330648863904, 136700965336, 37628617532, 8992149138, 1583635463, 249012444, 28541268, 2833808, 91997, 5449, -2242, 48, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 48*x^14 - 2242*x^13 + 5449*x^12 + 91997*x^11 + 2833808*x^10 + 28541268*x^9 + 249012444*x^8 + 1583635463*x^7 + 8992149138*x^6 + 37628617532*x^5 + 136700965336*x^4 + 330648863904*x^3 + 494720800128*x^2 + 444135049728*x + 190379307008)
 
gp: K = bnfinit(x^16 - 2*x^15 + 48*x^14 - 2242*x^13 + 5449*x^12 + 91997*x^11 + 2833808*x^10 + 28541268*x^9 + 249012444*x^8 + 1583635463*x^7 + 8992149138*x^6 + 37628617532*x^5 + 136700965336*x^4 + 330648863904*x^3 + 494720800128*x^2 + 444135049728*x + 190379307008, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 48 x^{14} - 2242 x^{13} + 5449 x^{12} + 91997 x^{11} + 2833808 x^{10} + 28541268 x^{9} + 249012444 x^{8} + 1583635463 x^{7} + 8992149138 x^{6} + 37628617532 x^{5} + 136700965336 x^{4} + 330648863904 x^{3} + 494720800128 x^{2} + 444135049728 x + 190379307008 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{16} a^{4} + \frac{7}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{3}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{128} a^{13} - \frac{1}{32} a^{12} + \frac{3}{64} a^{10} + \frac{5}{128} a^{9} + \frac{11}{128} a^{8} - \frac{3}{64} a^{7} + \frac{3}{16} a^{6} - \frac{3}{32} a^{5} + \frac{39}{128} a^{4} - \frac{5}{32} a^{3} + \frac{9}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{7936} a^{14} - \frac{9}{3968} a^{13} - \frac{7}{248} a^{12} + \frac{7}{3968} a^{11} + \frac{7}{256} a^{10} - \frac{163}{7936} a^{9} + \frac{9}{248} a^{8} - \frac{295}{1984} a^{7} - \frac{357}{1984} a^{6} - \frac{3593}{7936} a^{5} - \frac{1463}{3968} a^{4} + \frac{1}{64} a^{3} - \frac{7}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{165067405233817308800592738850069662025159910774174879111401803863702580359168} a^{15} - \frac{2115630086884282940491830177140417387546964426064114333801857555416535067}{82533702616908654400296369425034831012579955387087439555700901931851290179584} a^{14} - \frac{26308821081680015937728356099562114318297127927527292932489477482155961363}{20633425654227163600074092356258707753144988846771859888925225482962822544896} a^{13} + \frac{253023473674562195944256064458370932530977278453719408653215821459240278543}{82533702616908654400296369425034831012579955387087439555700901931851290179584} a^{12} + \frac{4032108824285782244732949556488398825387847706283683609990508644125801538353}{165067405233817308800592738850069662025159910774174879111401803863702580359168} a^{11} + \frac{9774461791370808044017947995845286288061839583219847808494387121371917482729}{165067405233817308800592738850069662025159910774174879111401803863702580359168} a^{10} + \frac{1633534588975749270792379968069325584287708374746912497808200912192824903519}{41266851308454327200148184712517415506289977693543719777850450965925645089792} a^{9} + \frac{35021669956666604313662744211312792452608505544847964972880606682872808057}{41266851308454327200148184712517415506289977693543719777850450965925645089792} a^{8} - \frac{68629853154683276946887433613068269804437868835501514353003095399572476087}{385671507555647917758394249649695472021401660687324483905144401550706963456} a^{7} - \frac{14288057545023599402199363599922063220423675771946575579947087262575011605097}{165067405233817308800592738850069662025159910774174879111401803863702580359168} a^{6} - \frac{40238470001417480514248931548380733224961627906421632319388077162918547117965}{82533702616908654400296369425034831012579955387087439555700901931851290179584} a^{5} - \frac{12615169023996095218737879647049796229676647945630060359525007485845886849647}{41266851308454327200148184712517415506289977693543719777850450965925645089792} a^{4} + \frac{20883587837696851658297189582609939811906266836258294018036384177190171487}{665594375942811729034648140524474443649838349895866448029845983321381372416} a^{3} - \frac{520823710459064155496492088013455126629259189296654322869501904774177603}{5199956062053216633083188597847456591014362108561456625233171744698291972} a^{2} + \frac{13973357543383327903674510711525575791462964255611456527891719639462638709}{41599648496425733064665508782779652728114896868491653001865373957586335776} a - \frac{799865189930489257964341405255533215252326762963178917868747388236973039}{5199956062053216633083188597847456591014362108561456625233171744698291972}$

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

Class group and class number

$C_{8}\times C_{320}$, which has order $2560$ (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:  \( 30203363305200 \) (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, 8.0.179859661768855001.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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