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Properties

Label 16.0.10602599994...1161.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{6}\cdot 37^{14}$
Root discriminant $48.88$
Ramified primes $7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21609, 68943, 99323, 980, -11557, -9835, -11836, 200, 2394, 21, 366, -106, 98, -34, 22, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 22*x^14 - 34*x^13 + 98*x^12 - 106*x^11 + 366*x^10 + 21*x^9 + 2394*x^8 + 200*x^7 - 11836*x^6 - 9835*x^5 - 11557*x^4 + 980*x^3 + 99323*x^2 + 68943*x + 21609)
 
gp: K = bnfinit(x^16 - 2*x^15 + 22*x^14 - 34*x^13 + 98*x^12 - 106*x^11 + 366*x^10 + 21*x^9 + 2394*x^8 + 200*x^7 - 11836*x^6 - 9835*x^5 - 11557*x^4 + 980*x^3 + 99323*x^2 + 68943*x + 21609, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 22 x^{14} - 34 x^{13} + 98 x^{12} - 106 x^{11} + 366 x^{10} + 21 x^{9} + 2394 x^{8} + 200 x^{7} - 11836 x^{6} - 9835 x^{5} - 11557 x^{4} + 980 x^{3} + 99323 x^{2} + 68943 x + 21609 \)

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:  \(1060259999412516731450111161=7^{6}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.88$
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:  $7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{189} a^{12} + \frac{26}{189} a^{11} - \frac{1}{7} a^{10} - \frac{2}{63} a^{9} - \frac{4}{27} a^{8} + \frac{2}{63} a^{7} + \frac{1}{21} a^{6} + \frac{5}{27} a^{5} - \frac{5}{27} a^{4} - \frac{29}{63} a^{3} + \frac{64}{189} a^{2} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{189} a^{13} - \frac{10}{189} a^{11} + \frac{1}{63} a^{10} + \frac{2}{189} a^{9} - \frac{22}{189} a^{8} - \frac{1}{9} a^{7} - \frac{10}{189} a^{6} + \frac{4}{189} a^{4} + \frac{58}{189} a^{3} - \frac{47}{189} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{181251} a^{14} + \frac{74}{60417} a^{13} - \frac{62}{181251} a^{12} + \frac{24403}{181251} a^{11} + \frac{185}{3699} a^{10} - \frac{9346}{181251} a^{9} - \frac{28082}{181251} a^{8} + \frac{3284}{25893} a^{7} - \frac{542}{8631} a^{6} + \frac{13430}{181251} a^{5} - \frac{19196}{60417} a^{4} + \frac{1201}{25893} a^{3} - \frac{10573}{25893} a^{2} - \frac{152}{1233} a - \frac{155}{411}$, $\frac{1}{2193284037966455337870903867} a^{15} + \frac{1493683444869311221531}{2193284037966455337870903867} a^{14} + \frac{53085822230972697500821}{243698226440717259763433763} a^{13} - \frac{105100736140438385364443}{2193284037966455337870903867} a^{12} + \frac{42470516429249773439778443}{313326291138065048267271981} a^{11} - \frac{124555671665216829555689417}{2193284037966455337870903867} a^{10} - \frac{94873068064724310673162772}{2193284037966455337870903867} a^{9} - \frac{3432446661142677786652096}{44760898734009292609610283} a^{8} - \frac{505383070181774173787456}{5911816813925755627684377} a^{7} - \frac{109798728712378735585903208}{731094679322151779290301289} a^{6} - \frac{40525418719835227795627061}{731094679322151779290301289} a^{5} - \frac{95649191294888374969273171}{313326291138065048267271981} a^{4} + \frac{27986028933454248930313706}{313326291138065048267271981} a^{3} + \frac{1704179176762895247594737}{14920299578003097536536761} a^{2} - \frac{1243947006556129527390520}{4973433192667699178845587} a + \frac{66424486842650326454317}{236830152031795198992647}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 17484605.3082 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

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

\(\Q(\sqrt{37}) \), 4.0.50653.1, 4.2.354571.1, 4.2.9583.1, 8.0.4651661979517.1 x2, 8.0.125720594041.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37Data not computed