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Properties

Label 16.0.11445391768...321.18
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 53^{12}$
Root discriminant $134.48$
Ramified primes $13, 53$
Class number $2048$ (GRH)
Class group $[2, 4, 4, 8, 8]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6828071, -16382503, 20319485, -16782028, 11390353, -6525662, 3249693, -1351737, 522505, -162303, 51702, -10176, 3119, -308, 93, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 93*x^14 - 308*x^13 + 3119*x^12 - 10176*x^11 + 51702*x^10 - 162303*x^9 + 522505*x^8 - 1351737*x^7 + 3249693*x^6 - 6525662*x^5 + 11390353*x^4 - 16782028*x^3 + 20319485*x^2 - 16382503*x + 6828071)
 
gp: K = bnfinit(x^16 - 4*x^15 + 93*x^14 - 308*x^13 + 3119*x^12 - 10176*x^11 + 51702*x^10 - 162303*x^9 + 522505*x^8 - 1351737*x^7 + 3249693*x^6 - 6525662*x^5 + 11390353*x^4 - 16782028*x^3 + 20319485*x^2 - 16382503*x + 6828071, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 93 x^{14} - 308 x^{13} + 3119 x^{12} - 10176 x^{11} + 51702 x^{10} - 162303 x^{9} + 522505 x^{8} - 1351737 x^{7} + 3249693 x^{6} - 6525662 x^{5} + 11390353 x^{4} - 16782028 x^{3} + 20319485 x^{2} - 16382503 x + 6828071 \)

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:  \(11445391768549949624817238631584321=13^{12}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.48$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{18648} a^{12} - \frac{1}{6216} a^{11} + \frac{599}{4662} a^{10} - \frac{101}{9324} a^{9} - \frac{143}{3108} a^{8} + \frac{5555}{18648} a^{7} - \frac{199}{18648} a^{6} - \frac{725}{9324} a^{5} - \frac{3349}{18648} a^{4} - \frac{159}{2072} a^{3} - \frac{7391}{18648} a^{2} - \frac{1465}{18648} a + \frac{3445}{18648}$, $\frac{1}{18648} a^{13} + \frac{341}{2664} a^{11} - \frac{167}{1332} a^{10} - \frac{61}{777} a^{9} + \frac{2981}{18648} a^{8} - \frac{1091}{9324} a^{7} + \frac{7277}{18648} a^{6} - \frac{7699}{18648} a^{5} - \frac{359}{3108} a^{4} - \frac{295}{2331} a^{3} + \frac{2167}{9324} a^{2} - \frac{475}{9324} a - \frac{2771}{6216}$, $\frac{1}{18648} a^{14} - \frac{643}{2664} a^{11} + \frac{1055}{4662} a^{10} + \frac{307}{18648} a^{9} + \frac{977}{4662} a^{8} - \frac{388}{2331} a^{7} + \frac{557}{9324} a^{6} - \frac{26}{2331} a^{5} - \frac{921}{2072} a^{4} - \frac{1777}{18648} a^{3} - \frac{8965}{18648} a^{2} - \frac{3929}{9324} a - \frac{1253}{2664}$, $\frac{1}{11379817578731583112282140696} a^{15} - \frac{27772355667575285851975}{2844954394682895778070535174} a^{14} - \frac{142312442462114368810399}{11379817578731583112282140696} a^{13} - \frac{39292537583328989201833}{3793272526243861037427380232} a^{12} - \frac{2410026716408922082326240899}{11379817578731583112282140696} a^{11} - \frac{2571019287695638336284626455}{11379817578731583112282140696} a^{10} + \frac{18547702494300691640395189}{474159065780482629678422529} a^{9} + \frac{239140188130007551272875469}{1264424175414620345809126744} a^{8} - \frac{115191447622243473507883207}{270948037588847216959098588} a^{7} + \frac{1968233970208223440974214615}{11379817578731583112282140696} a^{6} + \frac{130731526173555615314318293}{2844954394682895778070535174} a^{5} + \frac{369205143290947405258400323}{11379817578731583112282140696} a^{4} - \frac{118464975862323237929925523}{11379817578731583112282140696} a^{3} + \frac{1333489807713327116465212009}{5689908789365791556141070348} a^{2} + \frac{215841549054564865644565933}{1264424175414620345809126744} a + \frac{1306087104082233260313509339}{3793272526243861037427380232}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{8}\times C_{8}$, which has order $2048$ (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:  \( 8979144.94 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{689}) \), \(\Q(\sqrt{13}) \), 4.0.148877.1, \(\Q(\sqrt{13}, \sqrt{53})\), 4.0.25160213.2, 4.0.6171373.1 x2, 4.0.116441.1 x2, 8.0.633036318205369.2, 8.0.38085844705129.4, 8.8.106983137776707361.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$