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Properties

Label 16.0.13486335860...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{4}\cdot 109^{4}$
Root discriminant $156.90$
Ramified primes $2, 5, 29, 89, 109$
Class number $211328$ (GRH)
Class group $[2, 2, 2, 2, 13208]$ (GRH)
Galois group 16T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![192405821, 58641728, 95585948, 177951464, 28261438, -8533144, 19004161, 1562664, 784021, -58046, 119059, 958, 5229, -154, 97, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 97*x^14 - 154*x^13 + 5229*x^12 + 958*x^11 + 119059*x^10 - 58046*x^9 + 784021*x^8 + 1562664*x^7 + 19004161*x^6 - 8533144*x^5 + 28261438*x^4 + 177951464*x^3 + 95585948*x^2 + 58641728*x + 192405821)
 
gp: K = bnfinit(x^16 - 4*x^15 + 97*x^14 - 154*x^13 + 5229*x^12 + 958*x^11 + 119059*x^10 - 58046*x^9 + 784021*x^8 + 1562664*x^7 + 19004161*x^6 - 8533144*x^5 + 28261438*x^4 + 177951464*x^3 + 95585948*x^2 + 58641728*x + 192405821, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 97 x^{14} - 154 x^{13} + 5229 x^{12} + 958 x^{11} + 119059 x^{10} - 58046 x^{9} + 784021 x^{8} + 1562664 x^{7} + 19004161 x^{6} - 8533144 x^{5} + 28261438 x^{4} + 177951464 x^{3} + 95585948 x^{2} + 58641728 x + 192405821 \)

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:  \(134863358606854287590863897600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{4}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.90$
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:  $2, 5, 29, 89, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{4646956676438042874033863874904303033019171311930664765090405181313059} a^{15} - \frac{118544272963366761342275225681777022515078127824083050832941090947308}{4646956676438042874033863874904303033019171311930664765090405181313059} a^{14} - \frac{1763657637095484116075838413041901974072550221512320822425030249651009}{4646956676438042874033863874904303033019171311930664765090405181313059} a^{13} + \frac{482211808992870388243286325544672690583700663073276764504527515921283}{1548985558812680958011287958301434344339723770643554921696801727104353} a^{12} + \frac{1484927563829378013016997975250749706986626792863798038570146255404706}{4646956676438042874033863874904303033019171311930664765090405181313059} a^{11} + \frac{44017784382143887979716504323938182253358665171364377525272210320356}{1548985558812680958011287958301434344339723770643554921696801727104353} a^{10} + \frac{24630803103874438559484241760911152149146366152843699343888030666922}{1548985558812680958011287958301434344339723770643554921696801727104353} a^{9} + \frac{206122241729700135630385537323208826538283101303949060144773971797004}{663850953776863267719123410700614719002738758847237823584343597330437} a^{8} - \frac{6607663539994316942642513365111506273005114536253983783772305211745}{221283651258954422573041136900204906334246252949079274528114532443479} a^{7} - \frac{654339701242561374461433525253681745286220001883956252532004034562108}{1548985558812680958011287958301434344339723770643554921696801727104353} a^{6} + \frac{1055030441652662977544091976058943154950804370986820383216754157811118}{4646956676438042874033863874904303033019171311930664765090405181313059} a^{5} + \frac{38495595633415418952880807821710677871095913695574918010447142203161}{663850953776863267719123410700614719002738758847237823584343597330437} a^{4} - \frac{1566576824965975362180744852006312415928346628941491722591857456887920}{4646956676438042874033863874904303033019171311930664765090405181313059} a^{3} + \frac{264962102999552525892938528308312257530156861626124387708055312745862}{1548985558812680958011287958301434344339723770643554921696801727104353} a^{2} - \frac{527248185839776223564934433108377716112568468896864482276286633597397}{1548985558812680958011287958301434344339723770643554921696801727104353} a - \frac{947007293462795479093940752174235209212851177726937290237016150925136}{4646956676438042874033863874904303033019171311930664765090405181313059}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{13208}$, which has order $211328$ (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:  \( 245906.902496 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n839 are not computed
Character table for t16n839 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.1264400.2, 4.4.725.1, 4.4.43600.1, 8.8.1598707360000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
89Data not computed
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$