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Properties

Label 16.0.10249598790...4576.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 17^{12}$
Root discriminant $56.32$
Ramified primes $2, 17$
Class number $208$ (GRH)
Class group $[4, 52]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29207, -540, 48294, -6224, 56089, 8376, 9326, 2364, -914, -444, -34, -152, 77, 16, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 2*x^14 + 16*x^13 + 77*x^12 - 152*x^11 - 34*x^10 - 444*x^9 - 914*x^8 + 2364*x^7 + 9326*x^6 + 8376*x^5 + 56089*x^4 - 6224*x^3 + 48294*x^2 - 540*x + 29207)
 
gp: K = bnfinit(x^16 - 4*x^15 - 2*x^14 + 16*x^13 + 77*x^12 - 152*x^11 - 34*x^10 - 444*x^9 - 914*x^8 + 2364*x^7 + 9326*x^6 + 8376*x^5 + 56089*x^4 - 6224*x^3 + 48294*x^2 - 540*x + 29207, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 2 x^{14} + 16 x^{13} + 77 x^{12} - 152 x^{11} - 34 x^{10} - 444 x^{9} - 914 x^{8} + 2364 x^{7} + 9326 x^{6} + 8376 x^{5} + 56089 x^{4} - 6224 x^{3} + 48294 x^{2} - 540 x + 29207 \)

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:  \(10249598790959829536343064576=2^{44}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.32$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(272=2^{4}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{272}(123,·)$, $\chi_{272}(1,·)$, $\chi_{272}(67,·)$, $\chi_{272}(225,·)$, $\chi_{272}(137,·)$, $\chi_{272}(203,·)$, $\chi_{272}(81,·)$, $\chi_{272}(259,·)$, $\chi_{272}(217,·)$, $\chi_{272}(89,·)$, $\chi_{272}(33,·)$, $\chi_{272}(35,·)$, $\chi_{272}(169,·)$, $\chi_{272}(171,·)$, $\chi_{272}(115,·)$, $\chi_{272}(251,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{40} a^{12} - \frac{1}{5} a^{11} - \frac{1}{10} a^{10} - \frac{1}{2} a^{9} + \frac{3}{20} a^{8} - \frac{1}{5} a^{7} - \frac{1}{4} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{3}{10} a^{3} - \frac{1}{10} a^{2} - \frac{1}{5} a - \frac{7}{40}$, $\frac{1}{40} a^{13} + \frac{3}{10} a^{11} - \frac{3}{10} a^{10} + \frac{3}{20} a^{9} + \frac{3}{20} a^{7} - \frac{2}{5} a^{6} + \frac{3}{10} a^{4} - \frac{1}{2} a^{3} + \frac{9}{40} a - \frac{2}{5}$, $\frac{1}{988338057360} a^{14} - \frac{1433696701}{494169028680} a^{13} + \frac{4697409241}{988338057360} a^{12} + \frac{116096956573}{247084514340} a^{11} - \frac{49712979803}{494169028680} a^{10} + \frac{14453395127}{41180752390} a^{9} + \frac{13661537039}{49416902868} a^{8} - \frac{13880030621}{49416902868} a^{7} - \frac{79814521523}{164723009560} a^{6} + \frac{98961500627}{247084514340} a^{5} + \frac{32627990349}{82361504780} a^{4} + \frac{44551832993}{247084514340} a^{3} - \frac{170748044347}{988338057360} a^{2} - \frac{217320131293}{494169028680} a - \frac{67165051211}{988338057360}$, $\frac{1}{235406989268057120945520} a^{15} + \frac{3555621561}{39234498211342853490920} a^{14} + \frac{421754815710028346649}{78468996422685706981840} a^{13} + \frac{411004088136452859827}{39234498211342853490920} a^{12} + \frac{29894122580302476144089}{117703494634028560472760} a^{11} + \frac{8482852502342672317781}{29425873658507140118190} a^{10} + \frac{4023535731993780100951}{14712936829253570059095} a^{9} + \frac{4353997260370967953987}{14712936829253570059095} a^{8} + \frac{11419469061876802606229}{117703494634028560472760} a^{7} + \frac{6444825799722900105962}{14712936829253570059095} a^{6} - \frac{2699175063049430587453}{11770349463402856047276} a^{5} - \frac{15616121487760536888181}{58851747317014280236380} a^{4} + \frac{4556188424063757592523}{15693799284537141396368} a^{3} - \frac{1822155930647375073853}{23540698926805712094552} a^{2} + \frac{9003845640551240826381}{78468996422685706981840} a + \frac{58755173103144823153783}{117703494634028560472760}$

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

Class group and class number

$C_{4}\times C_{52}$, which has order $208$ (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:  \( 103646.40189541418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), 4.0.2048.2, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.591872.5, 4.0.10061824.2, 4.0.10061824.1, 4.4.314432.1, 4.4.4913.1, 8.0.350312464384.1, 8.0.101240302206976.1, 8.8.98867482624.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$