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Properties

Label 16.4.50744262454...8441.3
Degree $16$
Signature $[4, 6]$
Discriminant $11^{10}\cdot 89^{14}$
Root discriminant $227.29$
Ramified primes $11, 89$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-123169547, -493047953, 178813720, -99384084, -36249321, 43841512, -11659280, 767392, 304528, 18802, 6432, -2888, 2355, -359, 104, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 104*x^14 - 359*x^13 + 2355*x^12 - 2888*x^11 + 6432*x^10 + 18802*x^9 + 304528*x^8 + 767392*x^7 - 11659280*x^6 + 43841512*x^5 - 36249321*x^4 - 99384084*x^3 + 178813720*x^2 - 493047953*x - 123169547)
 
gp: K = bnfinit(x^16 - 6*x^15 + 104*x^14 - 359*x^13 + 2355*x^12 - 2888*x^11 + 6432*x^10 + 18802*x^9 + 304528*x^8 + 767392*x^7 - 11659280*x^6 + 43841512*x^5 - 36249321*x^4 - 99384084*x^3 + 178813720*x^2 - 493047953*x - 123169547, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 104 x^{14} - 359 x^{13} + 2355 x^{12} - 2888 x^{11} + 6432 x^{10} + 18802 x^{9} + 304528 x^{8} + 767392 x^{7} - 11659280 x^{6} + 43841512 x^{5} - 36249321 x^{4} - 99384084 x^{3} + 178813720 x^{2} - 493047953 x - 123169547 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(50744262454554467455972799262367678441=11^{10}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $227.29$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{704} a^{14} + \frac{7}{352} a^{13} - \frac{35}{704} a^{12} + \frac{27}{704} a^{11} - \frac{5}{88} a^{10} - \frac{41}{704} a^{9} + \frac{5}{176} a^{8} - \frac{139}{704} a^{7} + \frac{19}{88} a^{6} - \frac{287}{704} a^{5} + \frac{49}{176} a^{4} - \frac{347}{704} a^{3} - \frac{193}{704} a^{2} - \frac{111}{704} a + \frac{335}{704}$, $\frac{1}{251834046335963520950085289316822650310327396780573319746432} a^{15} + \frac{54404212550188488776169066238564551808742290851051658605}{251834046335963520950085289316822650310327396780573319746432} a^{14} - \frac{2869589605910933502520291696303597944099227778829039900689}{251834046335963520950085289316822650310327396780573319746432} a^{13} - \frac{5638079185962639837343500105891370123297208300439529974685}{125917023167981760475042644658411325155163698390286659873216} a^{12} - \frac{19178196632088198312208209202057385625825072513368158612011}{251834046335963520950085289316822650310327396780573319746432} a^{11} - \frac{6507075038061398116301056802274253157010461904320353069129}{251834046335963520950085289316822650310327396780573319746432} a^{10} + \frac{31891004871181346464482400419258641086212329641573144196693}{251834046335963520950085289316822650310327396780573319746432} a^{9} - \frac{1741402466510621828550654552130767296293329034224812976895}{251834046335963520950085289316822650310327396780573319746432} a^{8} - \frac{49401752047737554134648740000690107046451137475898347911517}{251834046335963520950085289316822650310327396780573319746432} a^{7} - \frac{8978056218685431258702951105509367894363620120839462524485}{22894004212360320086371389937892968210029763343688483613312} a^{6} + \frac{571702305106434999727951699301145275253572275602359508779}{251834046335963520950085289316822650310327396780573319746432} a^{5} + \frac{24622662572931024566415457842408061179883822549753788827241}{251834046335963520950085289316822650310327396780573319746432} a^{4} + \frac{50943022852177785090174429907039519685606695136316735881921}{125917023167981760475042644658411325155163698390286659873216} a^{3} + \frac{41164856962106363407589413524119749691120099859637214112313}{125917023167981760475042644658411325155163698390286659873216} a^{2} + \frac{2290577316735455990786695766784304078856264813671553673869}{11447002106180160043185694968946484105014881671844241806656} a + \frac{59666099330473308141011601914917936109411393698735440578857}{251834046335963520950085289316822650310327396780573319746432}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $9$
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:  \( 3035826822370 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.647590974205440089.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89Data not computed