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Properties

Label 16.16.1776912443...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{44}\cdot 5^{12}\cdot 11^{4}\cdot 41^{4}$
Root discriminant $103.66$
Ramified primes $2, 5, 11, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T203)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5487281519, 6665735720, 2787323020, -5441325612, 509655902, 1089455540, -214623836, -87322848, 20530069, 3540612, -891496, -77544, 19944, 876, -224, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 224*x^14 + 876*x^13 + 19944*x^12 - 77544*x^11 - 891496*x^10 + 3540612*x^9 + 20530069*x^8 - 87322848*x^7 - 214623836*x^6 + 1089455540*x^5 + 509655902*x^4 - 5441325612*x^3 + 2787323020*x^2 + 6665735720*x - 5487281519)
 
gp: K = bnfinit(x^16 - 4*x^15 - 224*x^14 + 876*x^13 + 19944*x^12 - 77544*x^11 - 891496*x^10 + 3540612*x^9 + 20530069*x^8 - 87322848*x^7 - 214623836*x^6 + 1089455540*x^5 + 509655902*x^4 - 5441325612*x^3 + 2787323020*x^2 + 6665735720*x - 5487281519, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 224 x^{14} + 876 x^{13} + 19944 x^{12} - 77544 x^{11} - 891496 x^{10} + 3540612 x^{9} + 20530069 x^{8} - 87322848 x^{7} - 214623836 x^{6} + 1089455540 x^{5} + 509655902 x^{4} - 5441325612 x^{3} + 2787323020 x^{2} + 6665735720 x - 5487281519 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(177691244381492740096000000000000=2^{44}\cdot 5^{12}\cdot 11^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.66$
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:  $2, 5, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{825} a^{14} + \frac{47}{825} a^{12} + \frac{404}{825} a^{11} + \frac{307}{825} a^{10} + \frac{1}{275} a^{9} + \frac{16}{275} a^{8} + \frac{74}{275} a^{7} - \frac{37}{165} a^{6} + \frac{124}{825} a^{5} + \frac{7}{55} a^{4} + \frac{3}{275} a^{3} - \frac{112}{825} a^{2} + \frac{179}{825} a - \frac{136}{825}$, $\frac{1}{42986354551601912069219882539477203904601618893911791932425} a^{15} + \frac{11208526936615898818803364818621786040916932468910840392}{42986354551601912069219882539477203904601618893911791932425} a^{14} + \frac{942071569926452140461976860570991872749055038515151471222}{42986354551601912069219882539477203904601618893911791932425} a^{13} + \frac{2423376471406559896865509487475986476996067150927978613778}{42986354551601912069219882539477203904601618893911791932425} a^{12} - \frac{142380805318279843043073185654536061507980898502082013813}{573151394021358827589598433859696052061354918585490559099} a^{11} - \frac{5017481223099686367564714765846924402588821557642975507778}{42986354551601912069219882539477203904601618893911791932425} a^{10} - \frac{2241212507094030278522196885910831884486608104585171195042}{14328784850533970689739960846492401301533872964637263977475} a^{9} + \frac{7028870727698493712158320498294545612706171680612906877821}{14328784850533970689739960846492401301533872964637263977475} a^{8} + \frac{19313082545494538228062414123988784254614458287393459718814}{42986354551601912069219882539477203904601618893911791932425} a^{7} + \frac{14962225050049039429511534324990384995016158917656849803754}{42986354551601912069219882539477203904601618893911791932425} a^{6} + \frac{75983788172164230108905338497727371896237481004416932962}{216011831917597548086532073062699517108550848713124582575} a^{5} + \frac{5873015887489049606673563743998657085252378418252915582923}{14328784850533970689739960846492401301533872964637263977475} a^{4} + \frac{12560440962340653939050208228059900845864388978024643880391}{42986354551601912069219882539477203904601618893911791932425} a^{3} - \frac{426653759545102887986793242681434328269571293915273768319}{1719454182064076482768795301579088156184064755756471677297} a^{2} + \frac{21122775598957470630650609587351585709914498968138749779207}{42986354551601912069219882539477203904601618893911791932425} a + \frac{2786525748653177772687851965414252121113419888761695646888}{42986354551601912069219882539477203904601618893911791932425}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 46111717836.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T203):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 41 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41Data not computed