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Properties

Label 16.0.85304566600...5184.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 449^{2}\cdot 1889^{5}$
Root discriminant $152.47$
Ramified primes $2, 449, 1889$
Class number $10762624$ (GRH)
Class group $[2, 2, 4, 672664]$ (GRH)
Galois group 16T1186

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![150100780000, 17326720000, 72875921600, -1069756000, 13706396484, -1007025840, 1413343656, -118159648, 81603792, -5940208, 2675448, -135296, 46253, -1308, 378, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 378*x^14 - 1308*x^13 + 46253*x^12 - 135296*x^11 + 2675448*x^10 - 5940208*x^9 + 81603792*x^8 - 118159648*x^7 + 1413343656*x^6 - 1007025840*x^5 + 13706396484*x^4 - 1069756000*x^3 + 72875921600*x^2 + 17326720000*x + 150100780000)
 
gp: K = bnfinit(x^16 - 4*x^15 + 378*x^14 - 1308*x^13 + 46253*x^12 - 135296*x^11 + 2675448*x^10 - 5940208*x^9 + 81603792*x^8 - 118159648*x^7 + 1413343656*x^6 - 1007025840*x^5 + 13706396484*x^4 - 1069756000*x^3 + 72875921600*x^2 + 17326720000*x + 150100780000, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 378 x^{14} - 1308 x^{13} + 46253 x^{12} - 135296 x^{11} + 2675448 x^{10} - 5940208 x^{9} + 81603792 x^{8} - 118159648 x^{7} + 1413343656 x^{6} - 1007025840 x^{5} + 13706396484 x^{4} - 1069756000 x^{3} + 72875921600 x^{2} + 17326720000 x + 150100780000 \)

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:  \(85304566600455667746642216756445184=2^{44}\cdot 449^{2}\cdot 1889^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.47$
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:  $2, 449, 1889$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{16} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{80} a^{13} + \frac{1}{80} a^{12} - \frac{1}{40} a^{11} + \frac{1}{40} a^{10} + \frac{3}{80} a^{9} + \frac{19}{80} a^{8} + \frac{1}{10} a^{7} + \frac{3}{20} a^{6} + \frac{1}{40} a^{5} + \frac{1}{40} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a$, $\frac{1}{800} a^{14} - \frac{1}{200} a^{13} - \frac{11}{400} a^{12} - \frac{1}{100} a^{11} + \frac{3}{800} a^{10} - \frac{3}{25} a^{9} - \frac{13}{200} a^{8} - \frac{27}{200} a^{7} - \frac{79}{400} a^{6} + \frac{11}{25} a^{5} + \frac{7}{100} a^{4} + \frac{9}{20} a^{3} + \frac{12}{25} a^{2}$, $\frac{1}{8889922282892208073589386034871509400775031182627617819875954708971484000} a^{15} + \frac{1550644920727862378316891284430499225198759527054717533536890735316643}{4444961141446104036794693017435754700387515591313808909937977354485742000} a^{14} - \frac{19646081388633819175924700048762327895760188494969000856151771975394391}{4444961141446104036794693017435754700387515591313808909937977354485742000} a^{13} + \frac{129533541545170318876650589952027911714857538974204498364675583108749031}{4444961141446104036794693017435754700387515591313808909937977354485742000} a^{12} + \frac{276421942836212073214911988273030176461629710289820594798292942674654683}{8889922282892208073589386034871509400775031182627617819875954708971484000} a^{11} - \frac{182946402874532954642791832673037152779145428148714639446339601266270963}{4444961141446104036794693017435754700387515591313808909937977354485742000} a^{10} + \frac{251784812375035927232143349653257398829498671808415209894544084647302027}{2222480570723052018397346508717877350193757795656904454968988677242871000} a^{9} + \frac{657956307135081007463799799307295791148401535405421396259645777786209981}{4444961141446104036794693017435754700387515591313808909937977354485742000} a^{8} + \frac{189526813893851418261634526078007250997111098600133017488186821795692361}{4444961141446104036794693017435754700387515591313808909937977354485742000} a^{7} + \frac{21090628657023902810963331364053882903391602683249837581647249806288883}{2222480570723052018397346508717877350193757795656904454968988677242871000} a^{6} + \frac{189458245889133612496194472117755142261988994405302477863815902511132467}{1111240285361526009198673254358938675096878897828452227484494338621435500} a^{5} + \frac{12930138002690311313544217722539889436528184604187139434118654920777579}{88899222828922080735893860348715094007750311826276178198759547089714840} a^{4} + \frac{75726506086227834386900897588437029983958838839914444727686004243028762}{277810071340381502299668313589734668774219724457113056871123584655358875} a^{3} + \frac{16964882488112927653863328997212260821851780199695974695144560237125567}{111124028536152600919867325435893867509687889782845222748449433862143550} a^{2} + \frac{3012101358493856267686253466130510595594283346179537811742911580039392}{11112402853615260091986732543589386750968788978284522274844943386214355} a + \frac{18749052701414381095844643234352511142228441636334356229295254393342}{2222480570723052018397346508717877350193757795656904454968988677242871}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{672664}$, which has order $10762624$ (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:  \( 56271.9156358 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1186:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 46 conjugacy class representatives for t16n1186
Character table for t16n1186 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
449Data not computed
1889Data not computed