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Properties

Label 16.0.18226469195...5744.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 3^{8}\cdot 7^{14}$
Root discriminant $15.99$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_8:C_2^2$ (as 16T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 54, 171, 315, 364, 301, 119, -9, -12, -52, 35, -7, 14, -7, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 35*x^10 - 52*x^9 - 12*x^8 - 9*x^7 + 119*x^6 + 301*x^5 + 364*x^4 + 315*x^3 + 171*x^2 + 54*x + 9)
 
gp: K = bnfinit(x^16 - 3*x^15 + 3*x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 35*x^10 - 52*x^9 - 12*x^8 - 9*x^7 + 119*x^6 + 301*x^5 + 364*x^4 + 315*x^3 + 171*x^2 + 54*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 3 x^{14} - 7 x^{13} + 14 x^{12} - 7 x^{11} + 35 x^{10} - 52 x^{9} - 12 x^{8} - 9 x^{7} + 119 x^{6} + 301 x^{5} + 364 x^{4} + 315 x^{3} + 171 x^{2} + 54 x + 9 \)

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:  \(18226469195621535744=2^{12}\cdot 3^{8}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.99$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{12} a^{8} - \frac{1}{2} a^{6} - \frac{1}{12} a^{5} + \frac{1}{12} a^{4} - \frac{1}{4} a^{3} + \frac{1}{12} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{13} + \frac{1}{6} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{5}{12} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{216} a^{14} + \frac{5}{216} a^{13} - \frac{1}{36} a^{11} - \frac{17}{108} a^{10} - \frac{7}{72} a^{9} + \frac{11}{72} a^{8} + \frac{35}{216} a^{7} - \frac{71}{216} a^{6} + \frac{13}{36} a^{5} - \frac{23}{54} a^{4} + \frac{11}{24} a^{3} - \frac{1}{3} a^{2} + \frac{1}{12} a - \frac{5}{24}$, $\frac{1}{41252735592} a^{15} + \frac{31150517}{20626367796} a^{14} - \frac{447468851}{41252735592} a^{13} + \frac{113481049}{3437727966} a^{12} - \frac{1108913333}{20626367796} a^{11} + \frac{8354902291}{41252735592} a^{10} - \frac{115394299}{2291818644} a^{9} - \frac{10110970187}{20626367796} a^{8} - \frac{1514980751}{5156591949} a^{7} + \frac{12170476667}{41252735592} a^{6} - \frac{1918420237}{5156591949} a^{5} - \frac{16366308175}{41252735592} a^{4} + \frac{2026709297}{4583637288} a^{3} + \frac{57285023}{572954661} a^{2} - \frac{436717279}{4583637288} a - \frac{1263785047}{4583637288}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{1039523}{7916472} a^{15} - \frac{11897951}{23749416} a^{14} + \frac{4653719}{5937354} a^{13} - \frac{164850}{109951} a^{12} + \frac{2961091}{989559} a^{11} - \frac{75869143}{23749416} a^{10} + \frac{54194945}{7916472} a^{9} - \frac{31995623}{2638824} a^{8} + \frac{177338813}{23749416} a^{7} - \frac{69913669}{11874708} a^{6} + \frac{79590245}{3958236} a^{5} + \frac{569641345}{23749416} a^{4} + \frac{32545709}{1319412} a^{3} + \frac{5444831}{329853} a^{2} + \frac{9002555}{2638824} a + \frac{1484351}{1319412} \) (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:  \( 4928.02836346 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T45):

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 $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), 4.0.1372.1, 4.0.12348.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.152473104.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$
7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$