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Properties

Label 16.0.61841248347...401.45
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{12}$
Root discriminant $149.44$
Ramified primes $13, 61$
Class number $147968$ (GRH)
Class group $[2, 2, 2, 2, 2, 68, 68]$ (GRH)
Galois group $Q_{16}$ (as 16T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17602483739, 10185930767, 7235716110, 3216880296, 1236318043, 424196239, 116312303, 28125680, 6562747, 996458, 268861, 19673, 8143, -118, 168, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 168*x^14 - 118*x^13 + 8143*x^12 + 19673*x^11 + 268861*x^10 + 996458*x^9 + 6562747*x^8 + 28125680*x^7 + 116312303*x^6 + 424196239*x^5 + 1236318043*x^4 + 3216880296*x^3 + 7235716110*x^2 + 10185930767*x + 17602483739)
 
gp: K = bnfinit(x^16 - 3*x^15 + 168*x^14 - 118*x^13 + 8143*x^12 + 19673*x^11 + 268861*x^10 + 996458*x^9 + 6562747*x^8 + 28125680*x^7 + 116312303*x^6 + 424196239*x^5 + 1236318043*x^4 + 3216880296*x^3 + 7235716110*x^2 + 10185930767*x + 17602483739, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 168 x^{14} - 118 x^{13} + 8143 x^{12} + 19673 x^{11} + 268861 x^{10} + 996458 x^{9} + 6562747 x^{8} + 28125680 x^{7} + 116312303 x^{6} + 424196239 x^{5} + 1236318043 x^{4} + 3216880296 x^{3} + 7235716110 x^{2} + 10185930767 x + 17602483739 \)

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:  \(61841248347955294332328765542314401=13^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $149.44$
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:  $13, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{15} + \frac{11738717040869622238855202182191879342592477993676984472006859916161434}{245128040875608074163846698458572389221885001276304981900567138836756141} a^{14} - \frac{55399416463338912894573479973785172816575609752891241282927022645975589}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{13} - \frac{253461479806437536549434250267026290281040428408026875573364898205583870}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{12} + \frac{29557294368606552883871828984670835151918593962265240163048517748148000}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{11} - \frac{116556598319913450171774318847526258638275166248887558340248049708269951}{245128040875608074163846698458572389221885001276304981900567138836756141} a^{10} + \frac{133385643843943504980907281456405878336862538620247935908718634794279413}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{9} - \frac{224447391321357525811748128705610981859324720160207122384872388488757235}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{8} + \frac{161124116958432768381142412411139223126598202384333504457325592109830137}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{7} - \frac{22974974312468788604836460785830853611069885635266673413351112451795255}{245128040875608074163846698458572389221885001276304981900567138836756141} a^{6} + \frac{64051286892088891407082081710180955308706041286826194969404279339480098}{245128040875608074163846698458572389221885001276304981900567138836756141} a^{5} + \frac{314049717561724651289645512470686673970527396981433026917254400798667994}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{4} - \frac{28436929454733189342691880621828775510621438269935493004205772017811261}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{3} - \frac{58225585842805866891410458426291455388538940519038221670880333877641931}{735384122626824222491540095375717167665655003828914945701701416510268423} a^{2} + \frac{325481855061790235376171771131609992771077258258729822032997639053346221}{735384122626824222491540095375717167665655003828914945701701416510268423} a + \frac{92772658612854696379564991708855293266736105440419665083137382770929541}{245128040875608074163846698458572389221885001276304981900567138836756141}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{68}\times C_{68}$, which has order $147968$ (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:  \( 233495.847112 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_{16}$ (as 16T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $Q_{16}$
Character table for $Q_{16}$

Intermediate fields

\(\Q(\sqrt{793}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{13}, \sqrt{61})\), 4.4.10309.1 x2, 4.4.48373.1 x2, 8.8.395451064801.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$