↑

Properties

Label 16.0.74963614410...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 11^{2}\cdot 61^{2}\cdot 101^{4}$
Root discriminant $47.83$
Ramified primes $2, 5, 11, 61, 101$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1161

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1021031, -1804726, 760059, 331560, 134818, -395902, 67733, 52332, 9097, -10446, 1977, 2070, 600, -64, -25, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 25*x^14 - 64*x^13 + 600*x^12 + 2070*x^11 + 1977*x^10 - 10446*x^9 + 9097*x^8 + 52332*x^7 + 67733*x^6 - 395902*x^5 + 134818*x^4 + 331560*x^3 + 760059*x^2 - 1804726*x + 1021031)
 
gp: K = bnfinit(x^16 - 4*x^15 - 25*x^14 - 64*x^13 + 600*x^12 + 2070*x^11 + 1977*x^10 - 10446*x^9 + 9097*x^8 + 52332*x^7 + 67733*x^6 - 395902*x^5 + 134818*x^4 + 331560*x^3 + 760059*x^2 - 1804726*x + 1021031, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 25 x^{14} - 64 x^{13} + 600 x^{12} + 2070 x^{11} + 1977 x^{10} - 10446 x^{9} + 9097 x^{8} + 52332 x^{7} + 67733 x^{6} - 395902 x^{5} + 134818 x^{4} + 331560 x^{3} + 760059 x^{2} - 1804726 x + 1021031 \)

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:  \(749636144106256000000000000=2^{16}\cdot 5^{12}\cdot 11^{2}\cdot 61^{2}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.83$
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:  $2, 5, 11, 61, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{145} a^{14} + \frac{1}{29} a^{13} - \frac{2}{145} a^{12} + \frac{8}{29} a^{11} - \frac{11}{145} a^{10} + \frac{47}{145} a^{9} + \frac{32}{145} a^{8} + \frac{31}{145} a^{7} + \frac{1}{145} a^{6} + \frac{68}{145} a^{5} - \frac{6}{29} a^{4} + \frac{9}{145} a^{3} - \frac{9}{29} a^{2} + \frac{67}{145} a + \frac{5}{29}$, $\frac{1}{54272633909686234601583538560415105715281743155} a^{15} + \frac{164144719478776856939263833894549951832356477}{54272633909686234601583538560415105715281743155} a^{14} - \frac{2616902870628384488882494963056025705230102887}{54272633909686234601583538560415105715281743155} a^{13} - \frac{5062812467437719919031329746652417127634518831}{54272633909686234601583538560415105715281743155} a^{12} + \frac{23819364620656817326352734535300785728160214758}{54272633909686234601583538560415105715281743155} a^{11} + \frac{10420483232955260226116577040089819994547658596}{54272633909686234601583538560415105715281743155} a^{10} + \frac{17650988653161594623069328190437104381957168069}{54272633909686234601583538560415105715281743155} a^{9} - \frac{22676607199169609271399640857830719531805475379}{54272633909686234601583538560415105715281743155} a^{8} + \frac{837692351012162737888937878917032921467297783}{1871470134816766710399432364152245024664887695} a^{7} - \frac{15862832949682940387907835534041569084989715937}{54272633909686234601583538560415105715281743155} a^{6} - \frac{20681104934779296036937731938734394739022491067}{54272633909686234601583538560415105715281743155} a^{5} - \frac{14925619307646447578105705216299380584314091762}{54272633909686234601583538560415105715281743155} a^{4} - \frac{11507706301929322280002226601161695222786270821}{54272633909686234601583538560415105715281743155} a^{3} + \frac{1015957687043074743862504125276309894966134327}{10854526781937246920316707712083021143056348631} a^{2} - \frac{696894996136476855243074964303828247792043845}{10854526781937246920316707712083021143056348631} a + \frac{20665436245263436374126991389800053285082624713}{54272633909686234601583538560415105715281743155}$

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

Class group and class number

$C_{4}$, 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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{238050904697801928043311332239907192}{279135703204150750660046692967762886141005} a^{15} - \frac{1224388610611769548636557591362541124}{279135703204150750660046692967762886141005} a^{14} - \frac{6145027860666103702717404645524361584}{279135703204150750660046692967762886141005} a^{13} - \frac{5120649926369789167444482886727482257}{279135703204150750660046692967762886141005} a^{12} + \frac{197716030223164435403370692675149524888}{279135703204150750660046692967762886141005} a^{11} + \frac{461094523676098251747906041517923312698}{279135703204150750660046692967762886141005} a^{10} - \frac{688929082964230523301560464044837347004}{279135703204150750660046692967762886141005} a^{9} - \frac{6544020928859358223914225095247953123331}{279135703204150750660046692967762886141005} a^{8} - \frac{2148509293178320394556107387632838000432}{279135703204150750660046692967762886141005} a^{7} + \frac{14012344942750997344020784062598689324312}{279135703204150750660046692967762886141005} a^{6} - \frac{3534529750651612762659924583652100120992}{279135703204150750660046692967762886141005} a^{5} - \frac{189589317861950100741597592447140469206547}{279135703204150750660046692967762886141005} a^{4} - \frac{29310288128360380591534191301262783749116}{279135703204150750660046692967762886141005} a^{3} + \frac{311475801482404423667963811446508526351514}{279135703204150750660046692967762886141005} a^{2} + \frac{373204447785130852763859094976120817192332}{279135703204150750660046692967762886141005} a - \frac{136737241866733943974160480358880115792381}{55827140640830150132009338593552577228201} \) (order $10$)
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:  \( 7589819.6836 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.404000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101Data not computed