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Properties

Label 16.0.76573533698...000.59
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $201.96$
Ramified primes $2, 3, 5, 19$
Class number $15458304$ (GRH)
Class group $[2, 2, 2, 4, 8, 60384]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10158336036, 3170793600, 2958375744, 679332960, 346620600, 64976400, 24649056, 3528000, 1084504, 114000, 39776, 4560, 1140, -120, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 - 120*x^13 + 1140*x^12 + 4560*x^11 + 39776*x^10 + 114000*x^9 + 1084504*x^8 + 3528000*x^7 + 24649056*x^6 + 64976400*x^5 + 346620600*x^4 + 679332960*x^3 + 2958375744*x^2 + 3170793600*x + 10158336036)
 
gp: K = bnfinit(x^16 - 16*x^14 - 120*x^13 + 1140*x^12 + 4560*x^11 + 39776*x^10 + 114000*x^9 + 1084504*x^8 + 3528000*x^7 + 24649056*x^6 + 64976400*x^5 + 346620600*x^4 + 679332960*x^3 + 2958375744*x^2 + 3170793600*x + 10158336036, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} - 120 x^{13} + 1140 x^{12} + 4560 x^{11} + 39776 x^{10} + 114000 x^{9} + 1084504 x^{8} + 3528000 x^{7} + 24649056 x^{6} + 64976400 x^{5} + 346620600 x^{4} + 679332960 x^{3} + 2958375744 x^{2} + 3170793600 x + 10158336036 \)

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:  \(7657353369870241665908736000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $201.96$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4560=2^{4}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4560}(1,·)$, $\chi_{4560}(3077,·)$, $\chi_{4560}(77,·)$, $\chi_{4560}(911,·)$, $\chi_{4560}(4369,·)$, $\chi_{4560}(1747,·)$, $\chi_{4560}(533,·)$, $\chi_{4560}(1369,·)$, $\chi_{4560}(2203,·)$, $\chi_{4560}(2279,·)$, $\chi_{4560}(2471,·)$, $\chi_{4560}(3307,·)$, $\chi_{4560}(3763,·)$, $\chi_{4560}(3001,·)$, $\chi_{4560}(3533,·)$, $\chi_{4560}(3839,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{24} a^{8} - \frac{1}{4} a^{4} - \frac{1}{6} a^{2} - \frac{1}{4}$, $\frac{1}{72} a^{9} - \frac{5}{12} a^{5} - \frac{1}{18} a^{3} - \frac{5}{12} a$, $\frac{1}{72} a^{10} - \frac{1}{12} a^{6} + \frac{5}{18} a^{4} - \frac{1}{12} a^{2}$, $\frac{1}{72} a^{11} - \frac{1}{12} a^{7} + \frac{5}{18} a^{5} - \frac{1}{12} a^{3}$, $\frac{1}{3240} a^{12} + \frac{1}{1620} a^{11} + \frac{1}{216} a^{10} + \frac{1}{540} a^{9} - \frac{1}{216} a^{8} - \frac{1}{54} a^{7} + \frac{109}{1620} a^{6} - \frac{31}{162} a^{5} + \frac{7}{54} a^{4} + \frac{83}{270} a^{3} + \frac{17}{36} a^{2} - \frac{23}{90} a - \frac{11}{60}$, $\frac{1}{3240} a^{13} + \frac{11}{3240} a^{11} + \frac{7}{1080} a^{10} + \frac{1}{180} a^{9} - \frac{1}{108} a^{8} + \frac{169}{1620} a^{7} - \frac{41}{540} a^{6} + \frac{31}{324} a^{5} - \frac{46}{135} a^{4} - \frac{107}{540} a^{3} + \frac{1}{20} a^{2} - \frac{4}{45} a + \frac{11}{30}$, $\frac{1}{49507753489200} a^{14} - \frac{32475623}{2750430749400} a^{13} + \frac{949522489}{8251292248200} a^{12} + \frac{7680739487}{12376938372300} a^{11} + \frac{13238698067}{4125646124100} a^{10} - \frac{39827012359}{8251292248200} a^{9} - \frac{323233820051}{24753876744600} a^{8} + \frac{21462808721}{687607687350} a^{7} + \frac{238981270913}{8251292248200} a^{6} + \frac{1891051889807}{6188469186150} a^{5} + \frac{3189007447}{38921189850} a^{4} + \frac{217647742751}{1031411531025} a^{3} - \frac{104110480067}{458405124900} a^{2} + \frac{516469758121}{1375215374700} a + \frac{90873943127}{458405124900}$, $\frac{1}{51717612891324269559075340729299728400} a^{15} + \frac{75354229231943409451}{40658500700726627011851682963285950} a^{14} + \frac{299749730880679374280110146643701}{5171761289132426955907534072929972840} a^{13} + \frac{28888928269107350761358491068419}{200455863919861509918896669493409800} a^{12} - \frac{18661309005047299842845004740474213}{8619602148554044926512556788216621400} a^{11} + \frac{10789800315579022002095770688736149}{1723920429710808985302511357643324280} a^{10} + \frac{179108568324107047430177723116986337}{25858806445662134779537670364649864200} a^{9} - \frac{113744625583300361559230115749884}{215490053713851123162813919705415535} a^{8} - \frac{151935005718901299945787549320729223}{1034352257826485391181506814585994568} a^{7} + \frac{231384402350548885486426732602441127}{4309801074277022463256278394108310700} a^{6} + \frac{3949365052543831862268605770427861}{28732007161846816421708522627388738} a^{5} - \frac{349351410761433450224113649121834593}{1436600358092340821085426131369436900} a^{4} + \frac{320404194117144172574853649813273}{26603710335043348538619002432767350} a^{3} - \frac{6166512242664942505690231749383507}{95773357206156054739028408757962460} a^{2} + \frac{177889175511181256483282107060713919}{478866786030780273695142043789812300} a + \frac{17723087726872932111783438347201827}{79811131005130045615857007298302050}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{8}\times C_{60384}$, which has order $15458304$ (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:  \( 2699811.4861512356 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-570}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}, \sqrt{-95})\), \(\Q(\sqrt{10}, \sqrt{-38})\), \(\Q(\sqrt{15}, \sqrt{-57})\), \(\Q(\sqrt{10}, \sqrt{-57})\), \(\Q(\sqrt{15}, \sqrt{-38})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{-38})\), 4.0.831744000.1, 4.4.2304000.2, 4.4.256000.1, 4.0.92416000.4, 8.0.432373800960000.158, 8.0.691798081536000000.15, 8.0.8540717056000000.25, 8.0.2767192326144000000.19, 8.0.2767192326144000000.18, 8.0.2767192326144000000.21, 8.8.21233664000000.3

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.24.13$x^{8} + 28 x^{4} + 36$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.13$x^{8} + 28 x^{4} + 36$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$