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Properties

Label 16.8.23150163082...5625.3
Degree $16$
Signature $[8, 4]$
Discriminant $5^{14}\cdot 41^{14}$
Root discriminant $105.39$
Ramified primes $5, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1115995, 500420, -15247495, -7132265, 8988535, 3584050, -996205, 39660, 138641, -12497, -15512, -2219, 1205, 166, -57, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 57*x^14 + 166*x^13 + 1205*x^12 - 2219*x^11 - 15512*x^10 - 12497*x^9 + 138641*x^8 + 39660*x^7 - 996205*x^6 + 3584050*x^5 + 8988535*x^4 - 7132265*x^3 - 15247495*x^2 + 500420*x + 1115995)
 
gp: K = bnfinit(x^16 - 2*x^15 - 57*x^14 + 166*x^13 + 1205*x^12 - 2219*x^11 - 15512*x^10 - 12497*x^9 + 138641*x^8 + 39660*x^7 - 996205*x^6 + 3584050*x^5 + 8988535*x^4 - 7132265*x^3 - 15247495*x^2 + 500420*x + 1115995, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 57 x^{14} + 166 x^{13} + 1205 x^{12} - 2219 x^{11} - 15512 x^{10} - 12497 x^{9} + 138641 x^{8} + 39660 x^{7} - 996205 x^{6} + 3584050 x^{5} + 8988535 x^{4} - 7132265 x^{3} - 15247495 x^{2} + 500420 x + 1115995 \)

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:  \(231501630828342033704595947265625=5^{14}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.39$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{542} a^{14} - \frac{42}{271} a^{13} + \frac{21}{271} a^{12} - \frac{205}{542} a^{11} - \frac{94}{271} a^{10} + \frac{39}{271} a^{9} + \frac{118}{271} a^{8} + \frac{121}{542} a^{7} + \frac{213}{542} a^{6} + \frac{175}{542} a^{5} + \frac{3}{542} a^{4} + \frac{89}{542} a^{3} - \frac{8}{271} a^{2} - \frac{21}{542} a - \frac{173}{542}$, $\frac{1}{33346011051613231151711714652560251543678101135261358} a^{15} + \frac{10033632628135428263543688549916229608324404348820}{16673005525806615575855857326280125771839050567630679} a^{14} - \frac{239089415174049306502267141000962028049575216873886}{16673005525806615575855857326280125771839050567630679} a^{13} + \frac{131168565984014310154513023864874650272580604052247}{16673005525806615575855857326280125771839050567630679} a^{12} - \frac{8334502907527434404512024464332713531259268670884043}{33346011051613231151711714652560251543678101135261358} a^{11} - \frac{1749653159171290364850311270377526004716497809462358}{16673005525806615575855857326280125771839050567630679} a^{10} - \frac{31479088334781985521793279683020180605730435044834}{16673005525806615575855857326280125771839050567630679} a^{9} + \frac{16337459570326937386069079028255422144799859458179301}{33346011051613231151711714652560251543678101135261358} a^{8} + \frac{4609598284218039517693014552561428982346123025293773}{33346011051613231151711714652560251543678101135261358} a^{7} - \frac{236812820142007253881012734901876253356437075957475}{33346011051613231151711714652560251543678101135261358} a^{6} + \frac{1056751401684274630796905443742314659366123225534772}{16673005525806615575855857326280125771839050567630679} a^{5} - \frac{3411773819969509325292253700141978453672961933698591}{33346011051613231151711714652560251543678101135261358} a^{4} + \frac{6162775688117938137255880322936230927705162590773085}{16673005525806615575855857326280125771839050567630679} a^{3} - \frac{7988705986559856466620346969599534673558727776704328}{16673005525806615575855857326280125771839050567630679} a^{2} + \frac{12191847915994857624525934589170690742213386200882441}{33346011051613231151711714652560251543678101135261358} a - \frac{264913430309602131452415727072747329162238448183837}{546655918878905428716585486107545107273411494020678}$

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

Class group and class number

$C_{4}$, 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:  $11$
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:  \( 4362739110.33 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 8.8.74220378765625.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$