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Properties

Label 16.0.71085478368...1921.4
Degree $16$
Signature $[0, 8]$
Discriminant $37^{4}\cdot 41^{14}$
Root discriminant $63.57$
Ramified primes $37, 41$
Class number $24$ (GRH)
Class group $[24]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![546239, 1302809, 2066545, 2128832, 1565210, 796163, 254629, 81298, 43027, 3597, 7039, -480, 670, -72, 38, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 38*x^14 - 72*x^13 + 670*x^12 - 480*x^11 + 7039*x^10 + 3597*x^9 + 43027*x^8 + 81298*x^7 + 254629*x^6 + 796163*x^5 + 1565210*x^4 + 2128832*x^3 + 2066545*x^2 + 1302809*x + 546239)
 
gp: K = bnfinit(x^16 - 2*x^15 + 38*x^14 - 72*x^13 + 670*x^12 - 480*x^11 + 7039*x^10 + 3597*x^9 + 43027*x^8 + 81298*x^7 + 254629*x^6 + 796163*x^5 + 1565210*x^4 + 2128832*x^3 + 2066545*x^2 + 1302809*x + 546239, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 38 x^{14} - 72 x^{13} + 670 x^{12} - 480 x^{11} + 7039 x^{10} + 3597 x^{9} + 43027 x^{8} + 81298 x^{7} + 254629 x^{6} + 796163 x^{5} + 1565210 x^{4} + 2128832 x^{3} + 2066545 x^{2} + 1302809 x + 546239 \)

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:  \(71085478368850138600216861921=37^{4}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.57$
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 a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{664} a^{13} + \frac{7}{166} a^{12} + \frac{1}{664} a^{11} - \frac{3}{332} a^{10} - \frac{65}{332} a^{9} - \frac{45}{332} a^{8} + \frac{131}{332} a^{7} + \frac{117}{664} a^{6} + \frac{25}{664} a^{5} - \frac{163}{664} a^{4} - \frac{71}{166} a^{3} + \frac{183}{664} a^{2} + \frac{65}{664} a - \frac{7}{83}$, $\frac{1}{1328} a^{14} - \frac{1}{1328} a^{13} - \frac{4}{83} a^{12} + \frac{297}{1328} a^{11} + \frac{127}{1328} a^{10} + \frac{263}{664} a^{9} + \frac{25}{664} a^{8} + \frac{321}{1328} a^{7} + \frac{225}{664} a^{6} - \frac{473}{1328} a^{5} + \frac{271}{664} a^{4} - \frac{231}{664} a^{3} - \frac{131}{664} a^{2} + \frac{79}{166} a + \frac{379}{1328}$, $\frac{1}{1001713481458537173440484037397344477136} a^{15} - \frac{149359503100940368402951538745417671}{500856740729268586720242018698672238568} a^{14} - \frac{326135896110449539145104598543981269}{1001713481458537173440484037397344477136} a^{13} + \frac{20092403010991395194385531748933872401}{1001713481458537173440484037397344477136} a^{12} + \frac{24567933381065723085898422900268741775}{125214185182317146680060504674668059642} a^{11} + \frac{80961751170704303687765411587451682039}{1001713481458537173440484037397344477136} a^{10} + \frac{20427764900876349193774825425850880607}{62607092591158573340030252337334029821} a^{9} - \frac{206582966180853709197818433139404437797}{1001713481458537173440484037397344477136} a^{8} - \frac{407835910270395437723938466816615331015}{1001713481458537173440484037397344477136} a^{7} + \frac{307119025687881287918203129712092887411}{1001713481458537173440484037397344477136} a^{6} - \frac{61457910573039931760617021397743766447}{1001713481458537173440484037397344477136} a^{5} - \frac{241155547876202187220901236593131947423}{500856740729268586720242018698672238568} a^{4} - \frac{19807851662335894929437442325790093631}{125214185182317146680060504674668059642} a^{3} - \frac{30891220585792591997310479471453766045}{125214185182317146680060504674668059642} a^{2} - \frac{289857414606882339735195643373583809407}{1001713481458537173440484037397344477136} a + \frac{380534266953750364751284079903506628897}{1001713481458537173440484037397344477136}$

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

Class group and class number

$C_{24}$, which has order $24$ (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:  \( 1905191.05667 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.2550077.1, 4.4.68921.1, 4.0.62197.1, 8.0.194754273881.1, 8.8.266618600943089.1, 8.0.6502892705929.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
$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}$