↑

Properties

Label 16.4.19718090210...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{8}\cdot 11^{6}\cdot 19^{8}$
Root discriminant $67.75$
Ramified primes $2, 5, 11, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T868

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2573195, -11893530, 14964706, -3711478, 1033292, 731524, -1014698, 775396, -325780, 128536, -28498, 7022, -457, 82, 34, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 34*x^14 + 82*x^13 - 457*x^12 + 7022*x^11 - 28498*x^10 + 128536*x^9 - 325780*x^8 + 775396*x^7 - 1014698*x^6 + 731524*x^5 + 1033292*x^4 - 3711478*x^3 + 14964706*x^2 - 11893530*x + 2573195)
 
gp: K = bnfinit(x^16 - 2*x^15 + 34*x^14 + 82*x^13 - 457*x^12 + 7022*x^11 - 28498*x^10 + 128536*x^9 - 325780*x^8 + 775396*x^7 - 1014698*x^6 + 731524*x^5 + 1033292*x^4 - 3711478*x^3 + 14964706*x^2 - 11893530*x + 2573195, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 34 x^{14} + 82 x^{13} - 457 x^{12} + 7022 x^{11} - 28498 x^{10} + 128536 x^{9} - 325780 x^{8} + 775396 x^{7} - 1014698 x^{6} + 731524 x^{5} + 1033292 x^{4} - 3711478 x^{3} + 14964706 x^{2} - 11893530 x + 2573195 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(197180902109852473753600000000=2^{24}\cdot 5^{8}\cdot 11^{6}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.75$
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:  $2, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{13} - \frac{2}{25} a^{12} + \frac{11}{25} a^{11} - \frac{12}{25} a^{9} + \frac{4}{25} a^{8} + \frac{1}{5} a^{7} - \frac{11}{25} a^{6} + \frac{7}{25} a^{5} - \frac{11}{25} a^{4} - \frac{3}{25} a^{3} - \frac{11}{25} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{2129570181441663637974335838512519098507417569934525} a^{15} - \frac{11926774006311181412353960619291003285507511783547}{2129570181441663637974335838512519098507417569934525} a^{14} + \frac{32563043151685399902484094597201589950548671071404}{425914036288332727594867167702503819701483513986905} a^{13} - \frac{197665689081486287522471452445468268637033556969441}{2129570181441663637974335838512519098507417569934525} a^{12} - \frac{728669155970064206727802565173452485580300867122749}{2129570181441663637974335838512519098507417569934525} a^{11} - \frac{316512084409463223017063667994006177763416801276892}{2129570181441663637974335838512519098507417569934525} a^{10} + \frac{45479720260127442668002988846594462495836480701331}{304224311634523376853476548358931299786773938562075} a^{9} - \frac{846303973837002569969753612583668072347302747337011}{2129570181441663637974335838512519098507417569934525} a^{8} - \frac{847401667734896211576735282081946031612602330853041}{2129570181441663637974335838512519098507417569934525} a^{7} + \frac{566249627681895353256392897796521002438733887537981}{2129570181441663637974335838512519098507417569934525} a^{6} - \frac{82425471583486979208866517690828245369537107198467}{304224311634523376853476548358931299786773938562075} a^{5} - \frac{160156804399523099265306627255662916745249875260804}{2129570181441663637974335838512519098507417569934525} a^{4} - \frac{838142693101954435753941447943129914563810690090424}{2129570181441663637974335838512519098507417569934525} a^{3} + \frac{620834666669562405215377058406321081599386478464629}{2129570181441663637974335838512519098507417569934525} a^{2} - \frac{23409938418401037134531170495631083732822902623746}{60844862326904675370695309671786259957354787712415} a - \frac{208381850179572683913694431160218919067875642433634}{425914036288332727594867167702503819701483513986905}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 110749388.664 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T868:

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 t16n868 are not computed
Character table for t16n868 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4400.1, 4.2.475.1, 4.2.83600.1, 8.4.6988960000.9

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.8.6.1$x^{8} + 57 x^{4} + 1444$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$