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Properties

Label 16.0.25833126466...5129.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 13^{14}$
Root discriminant $16.34$
Ramified primes $3, 13$
Class number $2$
Class group $[2]$
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 12, 22, 22, 6, -59, 107, -42, -20, 40, -17, -6, 15, -14, 9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 9*x^14 - 14*x^13 + 15*x^12 - 6*x^11 - 17*x^10 + 40*x^9 - 20*x^8 - 42*x^7 + 107*x^6 - 59*x^5 + 6*x^4 + 22*x^3 + 22*x^2 + 12*x + 9)
 
gp: K = bnfinit(x^16 - 4*x^15 + 9*x^14 - 14*x^13 + 15*x^12 - 6*x^11 - 17*x^10 + 40*x^9 - 20*x^8 - 42*x^7 + 107*x^6 - 59*x^5 + 6*x^4 + 22*x^3 + 22*x^2 + 12*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 9 x^{14} - 14 x^{13} + 15 x^{12} - 6 x^{11} - 17 x^{10} + 40 x^{9} - 20 x^{8} - 42 x^{7} + 107 x^{6} - 59 x^{5} + 6 x^{4} + 22 x^{3} + 22 x^{2} + 12 x + 9 \)

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:  \(25833126466573035129=3^{8}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.34$
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:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{81} a^{14} + \frac{1}{81} a^{13} - \frac{2}{81} a^{12} - \frac{13}{81} a^{11} + \frac{1}{9} a^{10} + \frac{4}{81} a^{9} - \frac{2}{27} a^{8} + \frac{1}{3} a^{7} + \frac{22}{81} a^{6} + \frac{14}{81} a^{5} - \frac{40}{81} a^{4} + \frac{1}{27} a^{3} + \frac{40}{81} a^{2} + \frac{4}{27} a + \frac{4}{9}$, $\frac{1}{810487053} a^{15} - \frac{1265666}{810487053} a^{14} + \frac{2715679}{810487053} a^{13} + \frac{33811118}{810487053} a^{12} + \frac{13010000}{270162351} a^{11} - \frac{62256254}{810487053} a^{10} - \frac{28762804}{270162351} a^{9} - \frac{38421100}{90054117} a^{8} + \frac{333927040}{810487053} a^{7} + \frac{64709783}{810487053} a^{6} - \frac{356280757}{810487053} a^{5} - \frac{1349408}{30018039} a^{4} + \frac{208509799}{810487053} a^{3} - \frac{129994927}{270162351} a^{2} - \frac{13204678}{90054117} a - \frac{261077}{30018039}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{9164740}{270162351} a^{15} + \frac{13084201}{90054117} a^{14} - \frac{31852996}{90054117} a^{13} + \frac{157439945}{270162351} a^{12} - \frac{186673942}{270162351} a^{11} + \frac{118364753}{270162351} a^{10} + \frac{112244704}{270162351} a^{9} - \frac{130498550}{90054117} a^{8} + \frac{304932836}{270162351} a^{7} + \frac{279186479}{270162351} a^{6} - \frac{347646325}{90054117} a^{5} + \frac{745848896}{270162351} a^{4} - \frac{298896988}{270162351} a^{3} - \frac{78555854}{270162351} a^{2} - \frac{140297417}{90054117} a + \frac{4875013}{30018039} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6428.85663404 \)
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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-3}) \), 4.0.2197.1, 4.4.19773.1, \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.564736653.1 x2, 8.4.5082629877.1 x2, 8.0.390971529.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$