↑

Properties

Label 16.8.13322563122...3281.3
Degree $16$
Signature $[8, 4]$
Discriminant $37^{8}\cdot 41^{14}$
Root discriminant $156.78$
Ramified primes $37, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40960000, 0, -258515968, 0, 118782528, 0, -3008, 0, -1228640, 0, -26685, 0, 2636, 0, 84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 84*x^14 + 2636*x^12 - 26685*x^10 - 1228640*x^8 - 3008*x^6 + 118782528*x^4 - 258515968*x^2 + 40960000)
 
gp: K = bnfinit(x^16 + 84*x^14 + 2636*x^12 - 26685*x^10 - 1228640*x^8 - 3008*x^6 + 118782528*x^4 - 258515968*x^2 + 40960000, 1)
 

Normalized defining polynomial

\( x^{16} + 84 x^{14} + 2636 x^{12} - 26685 x^{10} - 1228640 x^{8} - 3008 x^{6} + 118782528 x^{4} - 258515968 x^{2} + 40960000 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(133225631225242544609121034154723281=37^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.78$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{7} + \frac{1}{10} a^{5} + \frac{2}{5} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a$, $\frac{1}{20} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{7}{20} a^{2}$, $\frac{1}{200} a^{9} + \frac{1}{25} a^{7} + \frac{9}{50} a^{5} - \frac{9}{40} a^{3} - \frac{9}{50} a$, $\frac{1}{800} a^{10} - \frac{3}{200} a^{8} - \frac{1}{20} a^{7} - \frac{21}{200} a^{6} + \frac{1}{5} a^{5} - \frac{5}{32} a^{4} - \frac{1}{5} a^{3} + \frac{7}{25} a^{2} - \frac{7}{20} a$, $\frac{1}{1600} a^{11} - \frac{1}{400} a^{9} - \frac{1}{40} a^{8} - \frac{1}{80} a^{7} + \frac{1}{10} a^{6} + \frac{163}{1600} a^{5} - \frac{1}{10} a^{4} + \frac{83}{200} a^{3} + \frac{13}{40} a^{2} + \frac{8}{25} a$, $\frac{1}{1328000} a^{12} + \frac{1}{66400} a^{10} - \frac{1}{400} a^{9} - \frac{6117}{332000} a^{8} + \frac{3}{100} a^{7} - \frac{305213}{1328000} a^{6} + \frac{21}{100} a^{5} + \frac{3457}{41500} a^{4} - \frac{3}{16} a^{3} - \frac{9451}{41500} a^{2} - \frac{3}{50} a + \frac{29}{83}$, $\frac{1}{2656000} a^{13} + \frac{1}{132800} a^{11} + \frac{523}{664000} a^{9} - \frac{92733}{2656000} a^{7} + \frac{12587}{83000} a^{5} + \frac{36199}{83000} a^{3} - \frac{3613}{8300} a$, $\frac{1}{2544104228969747200000} a^{14} - \frac{1}{5312000} a^{13} + \frac{87415819467113}{636026057242436800000} a^{12} - \frac{1}{265600} a^{11} - \frac{28730652187089757}{636026057242436800000} a^{10} + \frac{2797}{1328000} a^{9} + \frac{45681637674449307011}{2544104228969747200000} a^{8} + \frac{198973}{5312000} a^{7} + \frac{30885457061514952713}{159006514310609200000} a^{6} + \frac{2353}{166000} a^{5} - \frac{9397631879222965951}{39751628577652300000} a^{4} + \frac{14063}{83000} a^{3} - \frac{3259326105356014991}{39751628577652300000} a^{2} + \frac{2119}{16600} a + \frac{320803604499671}{1987581428882615}$, $\frac{1}{20352833831757977600000} a^{15} + \frac{566351103535213}{5088208457939494400000} a^{13} - \frac{19151946505727757}{5088208457939494400000} a^{11} - \frac{1192716447863776189}{20352833831757977600000} a^{9} - \frac{5389878270903804653}{318013028621218400000} a^{7} - \frac{69688879049419802551}{318013028621218400000} a^{5} - \frac{67964440853524461191}{318013028621218400000} a^{3} - \frac{4727257325404059}{31801302862121840} a - \frac{1}{2}$

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:  $11$
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:  \( 2234156590970 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{1517}) \), \(\Q(\sqrt{37}) \), 4.4.94352849.1, 4.4.68921.1, \(\Q(\sqrt{37}, \sqrt{41})\), 8.4.365000864691088841.1 x2, 8.4.266618600943089.1 x2, 8.8.8902460114416801.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
37Data not computed
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$