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Properties

Label 16.8.50744262454...8441.1
Degree $16$
Signature $[8, 4]$
Discriminant $11^{10}\cdot 89^{14}$
Root discriminant $227.29$
Ramified primes $11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14685509, -60129950, 100836801, -80920028, 37615615, 7762122, -9360109, 4866906, -71786, -719664, 99979, 24370, -4365, 112, 4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 4365*x^12 + 24370*x^11 + 99979*x^10 - 719664*x^9 - 71786*x^8 + 4866906*x^7 - 9360109*x^6 + 7762122*x^5 + 37615615*x^4 - 80920028*x^3 + 100836801*x^2 - 60129950*x + 14685509)
 
gp: K = bnfinit(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 4365*x^12 + 24370*x^11 + 99979*x^10 - 719664*x^9 - 71786*x^8 + 4866906*x^7 - 9360109*x^6 + 7762122*x^5 + 37615615*x^4 - 80920028*x^3 + 100836801*x^2 - 60129950*x + 14685509, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 4 x^{14} + 112 x^{13} - 4365 x^{12} + 24370 x^{11} + 99979 x^{10} - 719664 x^{9} - 71786 x^{8} + 4866906 x^{7} - 9360109 x^{6} + 7762122 x^{5} + 37615615 x^{4} - 80920028 x^{3} + 100836801 x^{2} - 60129950 x + 14685509 \)

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:  \(50744262454554467455972799262367678441=11^{10}\cdot 89^{14}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{8} - \frac{2}{11} a^{7} + \frac{3}{22} a^{6} + \frac{5}{22} a^{5} - \frac{4}{11} a^{4} + \frac{3}{22} a^{3} - \frac{9}{22} a^{2} + \frac{9}{22} a + \frac{9}{22}$, $\frac{1}{22} a^{9} - \frac{1}{11} a^{7} - \frac{5}{22} a^{6} - \frac{5}{11} a^{5} - \frac{7}{22} a^{4} - \frac{4}{11} a^{3} - \frac{5}{22} a^{2} - \frac{5}{11} a + \frac{3}{22}$, $\frac{1}{242} a^{10} - \frac{5}{242} a^{9} - \frac{1}{242} a^{8} + \frac{17}{121} a^{7} - \frac{59}{242} a^{6} + \frac{37}{242} a^{5} + \frac{48}{121} a^{4} + \frac{19}{121} a^{3} + \frac{58}{121} a^{2} - \frac{15}{242} a + \frac{41}{121}$, $\frac{1}{242} a^{11} - \frac{2}{121} a^{9} - \frac{2}{121} a^{8} - \frac{43}{242} a^{7} + \frac{17}{242} a^{6} - \frac{52}{121} a^{5} - \frac{49}{121} a^{4} + \frac{31}{242} a^{3} + \frac{13}{121} a^{2} - \frac{13}{121} a - \frac{63}{242}$, $\frac{1}{484} a^{12} - \frac{1}{484} a^{10} - \frac{2}{121} a^{9} + \frac{9}{484} a^{8} - \frac{1}{242} a^{7} + \frac{71}{484} a^{6} - \frac{16}{121} a^{5} - \frac{9}{22} a^{4} + \frac{24}{121} a^{3} + \frac{7}{242} a^{2} + \frac{39}{121} a - \frac{73}{484}$, $\frac{1}{484} a^{13} - \frac{1}{484} a^{11} - \frac{9}{484} a^{9} - \frac{5}{242} a^{8} + \frac{57}{484} a^{7} + \frac{20}{121} a^{6} + \frac{30}{121} a^{5} - \frac{4}{121} a^{4} + \frac{71}{242} a^{3} + \frac{3}{242} a^{2} - \frac{171}{484} a + \frac{119}{242}$, $\frac{1}{8256895281847128865696} a^{14} - \frac{7}{8256895281847128865696} a^{13} - \frac{644810925229930059}{2064223820461782216424} a^{12} + \frac{15475462205518321507}{8256895281847128865696} a^{11} - \frac{1798659982489363167}{4128447640923564432848} a^{10} - \frac{123871803725690982311}{8256895281847128865696} a^{9} - \frac{67227789868516333303}{4128447640923564432848} a^{8} + \frac{19880592130999699961}{113108154545851080352} a^{7} - \frac{630394576115209221467}{8256895281847128865696} a^{6} - \frac{1939399610516699669109}{4128447640923564432848} a^{5} - \frac{740278853492737681029}{2064223820461782216424} a^{4} - \frac{65652248582025527283}{258027977557722777053} a^{3} - \frac{438621358442379738029}{8256895281847128865696} a^{2} + \frac{50059318336341155093}{750626843804284442336} a - \frac{362499595124309679827}{8256895281847128865696}$, $\frac{1}{284920685490698875768571872} a^{15} + \frac{8623}{142460342745349437884285936} a^{14} + \frac{59655195137744141405561}{284920685490698875768571872} a^{13} - \frac{137920198462400276061889}{284920685490698875768571872} a^{12} - \frac{378834548476766380800471}{284920685490698875768571872} a^{11} - \frac{44191358472543026839279}{25901880499154443251688352} a^{10} - \frac{272446905338540836747755}{25901880499154443251688352} a^{9} - \frac{111584386710626740835735}{25901880499154443251688352} a^{8} + \frac{974334877522835278621163}{12950940249577221625844176} a^{7} + \frac{6274393976726350689539157}{25901880499154443251688352} a^{6} + \frac{3625702554001707981616791}{12950940249577221625844176} a^{5} + \frac{22115760557076494972646767}{71230171372674718942142968} a^{4} + \frac{71795545293519050913016163}{284920685490698875768571872} a^{3} + \frac{25053407874880838695144531}{142460342745349437884285936} a^{2} - \frac{11831274644019819635393311}{35615085686337359471071484} a - \frac{112788745709609308439940439}{284920685490698875768571872}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 13422288629500 \) (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$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.3.2$x^{4} - 11$$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