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Properties

Label 16.0.68372792983...489.17
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 67^{8}$
Root discriminant $97.65$
Ramified primes $17, 67$
Class number $1152$ (GRH)
Class group $[2, 4, 12, 12]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11810851, 6638631, 4896552, 2547293, 1612872, 359124, -124811, -5118, 58610, 10007, -5264, -488, 753, 6, -38, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 38*x^14 + 6*x^13 + 753*x^12 - 488*x^11 - 5264*x^10 + 10007*x^9 + 58610*x^8 - 5118*x^7 - 124811*x^6 + 359124*x^5 + 1612872*x^4 + 2547293*x^3 + 4896552*x^2 + 6638631*x + 11810851)
 
gp: K = bnfinit(x^16 - 2*x^15 - 38*x^14 + 6*x^13 + 753*x^12 - 488*x^11 - 5264*x^10 + 10007*x^9 + 58610*x^8 - 5118*x^7 - 124811*x^6 + 359124*x^5 + 1612872*x^4 + 2547293*x^3 + 4896552*x^2 + 6638631*x + 11810851, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 38 x^{14} + 6 x^{13} + 753 x^{12} - 488 x^{11} - 5264 x^{10} + 10007 x^{9} + 58610 x^{8} - 5118 x^{7} - 124811 x^{6} + 359124 x^{5} + 1612872 x^{4} + 2547293 x^{3} + 4896552 x^{2} + 6638631 x + 11810851 \)

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:  \(68372792983010839504523425519489=17^{14}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.65$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{26} a^{14} + \frac{1}{13} a^{13} + \frac{3}{13} a^{12} + \frac{11}{26} a^{11} + \frac{6}{13} a^{10} + \frac{4}{13} a^{9} - \frac{3}{26} a^{8} + \frac{1}{13} a^{6} - \frac{1}{26} a^{5} - \frac{4}{13} a^{4} - \frac{2}{13} a^{3} + \frac{9}{26} a^{2} + \frac{2}{13} a$, $\frac{1}{1920040691911840395830554291177909184551110610704398} a^{15} - \frac{5752850821579089214499311811174921140310743181230}{960020345955920197915277145588954592275555305352199} a^{14} + \frac{52413611710660044380649287977825168712570412979937}{960020345955920197915277145588954592275555305352199} a^{13} - \frac{136006764190068548818334777374797686830934360035359}{1920040691911840395830554291177909184551110610704398} a^{12} - \frac{91460248590304309404203728568674336158561116122250}{960020345955920197915277145588954592275555305352199} a^{11} - \frac{22976772974344117350892264111439846801447809432125}{960020345955920197915277145588954592275555305352199} a^{10} - \frac{617810071519161188273066472870302363404793060947309}{1920040691911840395830554291177909184551110610704398} a^{9} - \frac{121495594560451519120809388558073630141683233026643}{960020345955920197915277145588954592275555305352199} a^{8} - \frac{286763878799595568062863064785633252539916807780587}{960020345955920197915277145588954592275555305352199} a^{7} + \frac{492307119327474479934433608932969476343008905828841}{1920040691911840395830554291177909184551110610704398} a^{6} + \frac{18800201469817412153667058797016322197443016935837}{960020345955920197915277145588954592275555305352199} a^{5} - \frac{376819656409133470552984201747285958075263968217995}{960020345955920197915277145588954592275555305352199} a^{4} + \frac{24875051964832461777507683644053475413290903477903}{147695437839372338140811868552146860350085431592646} a^{3} - \frac{180048288110273268006381655518258651890799103462476}{960020345955920197915277145588954592275555305352199} a^{2} - \frac{136830935332717094969974273992345483937771081815341}{960020345955920197915277145588954592275555305352199} a + \frac{28384305989533074644506368986840106579056299967204}{73847718919686169070405934276073430175042715796323}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{12}\times C_{12}$, which has order $1152$ (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:  \( 1426256.46809 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.2.27492691091.1, 8.0.8268784250602433.5, 8.6.7259687665147.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$67$67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$