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Properties

Label 16.8.42010250796...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{4}\cdot 3^{8}\cdot 5^{10}\cdot 7^{8}\cdot 37^{4}\cdot 41^{14}$
Root discriminant $947.24$
Ramified primes $2, 3, 5, 7, 37, 41$
Class number $1024$ (GRH)
Class group $[2, 2, 4, 8, 8]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1284521964547264, 391275792985264, -221066965507360, -5131102198184, 9840293716774, -1648592706546, 185224572711, -31134488218, 4325628545, -127119113, -31999820, 3821262, -95772, 2893, -511, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 511*x^14 + 2893*x^13 - 95772*x^12 + 3821262*x^11 - 31999820*x^10 - 127119113*x^9 + 4325628545*x^8 - 31134488218*x^7 + 185224572711*x^6 - 1648592706546*x^5 + 9840293716774*x^4 - 5131102198184*x^3 - 221066965507360*x^2 + 391275792985264*x + 1284521964547264)
 
gp: K = bnfinit(x^16 - 6*x^15 - 511*x^14 + 2893*x^13 - 95772*x^12 + 3821262*x^11 - 31999820*x^10 - 127119113*x^9 + 4325628545*x^8 - 31134488218*x^7 + 185224572711*x^6 - 1648592706546*x^5 + 9840293716774*x^4 - 5131102198184*x^3 - 221066965507360*x^2 + 391275792985264*x + 1284521964547264, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 511 x^{14} + 2893 x^{13} - 95772 x^{12} + 3821262 x^{11} - 31999820 x^{10} - 127119113 x^{9} + 4325628545 x^{8} - 31134488218 x^{7} + 185224572711 x^{6} - 1648592706546 x^{5} + 9840293716774 x^{4} - 5131102198184 x^{3} - 221066965507360 x^{2} + 391275792985264 x + 1284521964547264 \)

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:  \(420102507961629136806462464459178014450156250000=2^{4}\cdot 3^{8}\cdot 5^{10}\cdot 7^{8}\cdot 37^{4}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $947.24$
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:  $2, 3, 5, 7, 37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{41} a^{8} - \frac{3}{41} a^{7} - \frac{14}{41} a^{6} - \frac{10}{41} a^{5} - \frac{3}{41} a^{4} + \frac{5}{41} a^{3} + \frac{17}{41} a^{2} - \frac{15}{41} a + \frac{18}{41}$, $\frac{1}{82} a^{9} - \frac{1}{82} a^{8} + \frac{21}{82} a^{7} - \frac{19}{41} a^{6} + \frac{9}{41} a^{5} - \frac{1}{82} a^{4} + \frac{27}{82} a^{3} + \frac{19}{82} a^{2} - \frac{6}{41} a + \frac{18}{41}$, $\frac{1}{82} a^{10} - \frac{39}{82} a^{7} + \frac{7}{41} a^{6} - \frac{29}{82} a^{5} + \frac{2}{41} a^{4} + \frac{14}{41} a^{3} - \frac{5}{82} a^{2} - \frac{2}{41} a + \frac{2}{41}$, $\frac{1}{164} a^{11} + \frac{1}{164} a^{8} + \frac{29}{82} a^{7} - \frac{15}{164} a^{6} - \frac{17}{41} a^{5} - \frac{5}{82} a^{4} - \frac{51}{164} a^{3} - \frac{31}{82} a^{2} - \frac{11}{82} a + \frac{16}{41}$, $\frac{1}{12136} a^{12} - \frac{3}{1517} a^{11} + \frac{7}{1517} a^{10} - \frac{3}{12136} a^{9} + \frac{21}{6068} a^{8} + \frac{3037}{12136} a^{7} - \frac{571}{1517} a^{6} + \frac{517}{6068} a^{5} + \frac{405}{12136} a^{4} + \frac{337}{6068} a^{3} + \frac{1491}{6068} a^{2} + \frac{229}{1517} a + \frac{319}{1517}$, $\frac{1}{12136} a^{13} - \frac{1}{6068} a^{11} + \frac{9}{12136} a^{10} - \frac{15}{6068} a^{9} + \frac{3}{296} a^{8} + \frac{733}{1517} a^{7} - \frac{5}{1517} a^{6} + \frac{209}{12136} a^{5} + \frac{535}{6068} a^{4} + \frac{1515}{3034} a^{3} - \frac{408}{1517} a^{2} - \frac{543}{3034} a - \frac{373}{1517}$, $\frac{1}{3762160} a^{14} - \frac{147}{3762160} a^{13} - \frac{49}{1881080} a^{12} + \frac{5789}{3762160} a^{11} + \frac{19171}{3762160} a^{10} + \frac{1201}{752432} a^{9} + \frac{29261}{3762160} a^{8} - \frac{75689}{235135} a^{7} - \frac{105149}{752432} a^{6} - \frac{1461657}{3762160} a^{5} - \frac{349541}{1881080} a^{4} - \frac{10973}{60680} a^{3} - \frac{10689}{25420} a^{2} - \frac{84713}{235135} a - \frac{33129}{235135}$, $\frac{1}{697588734859458426616016860282017963410691804175683751516311451157479731743497842477409600} a^{15} - \frac{42916573999864069888491512618411345950243031033563863237653709720907807397044114399}{697588734859458426616016860282017963410691804175683751516311451157479731743497842477409600} a^{14} + \frac{5461034003610518348527163681780900617807115199815290922358109072657517184184769546949}{174397183714864606654004215070504490852672951043920937879077862789369932935874460619352400} a^{13} + \frac{1932709044862652271945379598649473766278932625698051419278651947159470697399238131673}{139517746971891685323203372056403592682138360835136750303262290231495946348699568495481920} a^{12} + \frac{339032693493640065973226994729493782821165794937172179826293750850048082414573650819383}{697588734859458426616016860282017963410691804175683751516311451157479731743497842477409600} a^{11} - \frac{2061892529183250227523419120918960617969569858140332806566053542923639689927771200126357}{697588734859458426616016860282017963410691804175683751516311451157479731743497842477409600} a^{10} - \frac{823494794811998528687864126228346162870783611877211205518489331876374360650211981472819}{697588734859458426616016860282017963410691804175683751516311451157479731743497842477409600} a^{9} - \frac{1691867764989870687450936799086227574115935152032040024150251223016626344455386058550673}{348794367429729213308008430141008981705345902087841875758155725578739865871748921238704800} a^{8} - \frac{4215518030948491466887790500119845782075691137491165429051683595462348578905357124613003}{11823537878973871637559607801390134973062572952130233076547651714533554775313522753854400} a^{7} + \frac{12698795278691740118426488559256744661295846613785704965632355247520810589966207453501543}{697588734859458426616016860282017963410691804175683751516311451157479731743497842477409600} a^{6} - \frac{86723924285572653066437265406327997847844490230686908820335648798937033133348407192952097}{174397183714864606654004215070504490852672951043920937879077862789369932935874460619352400} a^{5} - \frac{139622523798434001462211896045066522033434605957670108492494778878256036076672643856321481}{348794367429729213308008430141008981705345902087841875758155725578739865871748921238704800} a^{4} + \frac{956808294972994141870904291253494783532092633651380364553616566938486543474547561318879}{4359929592871615166350105376762612271316823776098023446976946569734248323396861515483810} a^{3} - \frac{167894570118080233079998719594127909126734347680807938944901269548238238075787008413382}{473905390529523387646750584430718725143133019141089505106189844536331339499658860378675} a^{2} - \frac{15751099171567083489085299103396394176455100263345751794929770458027753856023585843435093}{43599295928716151663501053767626122713168237760980234469769465697342483233968615154838100} a + \frac{2607230858098662054914231745053213744325830677670158342185742099629837145101390367722057}{10899823982179037915875263441906530678292059440245058617442366424335620808492153788709525}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{8}\times C_{8}$, which has order $1024$ (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:  \( 40415348358700000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{4305}) \), \(\Q(\sqrt{105}) \), 4.4.3101445.1, 4.4.16885645.1, \(\Q(\sqrt{41}, \sqrt{105})\), 8.8.577378139308700625.1

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

Sibling fields

Degree 16 siblings: 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 R R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41Data not computed