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Properties

Label 16.0.28596554320...5089.4
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 19^{8}$
Root discriminant $52.00$
Ramified primes $17, 19$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![630208, -1857456, 1746316, -607918, 262361, -242978, 129750, -26056, 4010, -4202, 2102, -158, -102, -8, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 8*x^13 - 102*x^12 - 158*x^11 + 2102*x^10 - 4202*x^9 + 4010*x^8 - 26056*x^7 + 129750*x^6 - 242978*x^5 + 262361*x^4 - 607918*x^3 + 1746316*x^2 - 1857456*x + 630208)
 
gp: K = bnfinit(x^16 - 8*x^15 + 26*x^14 - 8*x^13 - 102*x^12 - 158*x^11 + 2102*x^10 - 4202*x^9 + 4010*x^8 - 26056*x^7 + 129750*x^6 - 242978*x^5 + 262361*x^4 - 607918*x^3 + 1746316*x^2 - 1857456*x + 630208, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 26 x^{14} - 8 x^{13} - 102 x^{12} - 158 x^{11} + 2102 x^{10} - 4202 x^{9} + 4010 x^{8} - 26056 x^{7} + 129750 x^{6} - 242978 x^{5} + 262361 x^{4} - 607918 x^{3} + 1746316 x^{2} - 1857456 x + 630208 \)

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:  \(2859655432078149808865465089=17^{14}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.00$
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:  $17, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{104} a^{11} + \frac{1}{52} a^{10} + \frac{1}{52} a^{9} - \frac{3}{26} a^{8} - \frac{3}{26} a^{7} - \frac{3}{13} a^{6} + \frac{23}{104} a^{5} - \frac{3}{52} a^{4} + \frac{25}{52} a^{3} - \frac{2}{13} a^{2} + \frac{1}{26} a - \frac{4}{13}$, $\frac{1}{3536} a^{12} + \frac{3}{1768} a^{11} + \frac{31}{1768} a^{10} + \frac{191}{3536} a^{9} - \frac{1}{104} a^{8} - \frac{61}{442} a^{7} + \frac{811}{3536} a^{6} + \frac{251}{1768} a^{5} - \frac{5}{136} a^{4} - \frac{1155}{3536} a^{3} - \frac{745}{1768} a^{2} - \frac{251}{884} a - \frac{21}{221}$, $\frac{1}{3536} a^{13} - \frac{1}{442} a^{11} + \frac{193}{3536} a^{10} + \frac{3}{136} a^{9} + \frac{31}{884} a^{8} + \frac{47}{272} a^{7} - \frac{3}{884} a^{6} - \frac{97}{884} a^{5} - \frac{613}{3536} a^{4} - \frac{33}{104} a^{3} + \frac{88}{221} a^{2} - \frac{95}{221} a - \frac{27}{221}$, $\frac{1}{41668224} a^{14} + \frac{1127}{41668224} a^{13} + \frac{2719}{20834112} a^{12} + \frac{180287}{41668224} a^{11} - \frac{1212827}{41668224} a^{10} - \frac{214951}{20834112} a^{9} + \frac{3770947}{41668224} a^{8} + \frac{3423}{1068416} a^{7} + \frac{731283}{6944704} a^{6} + \frac{315505}{2451072} a^{5} + \frac{591911}{3205248} a^{4} + \frac{6416167}{20834112} a^{3} - \frac{1468575}{3472352} a^{2} - \frac{247727}{2604264} a + \frac{220565}{651066}$, $\frac{1}{484936125682428181632} a^{15} + \frac{203311911697}{484936125682428181632} a^{14} - \frac{276643483351881}{2377137870992295008} a^{13} - \frac{9752873218143031}{161645375227476060544} a^{12} - \frac{2104768407855226645}{484936125682428181632} a^{11} + \frac{1683869711972893035}{40411343806869015136} a^{10} + \frac{3914977547585689885}{161645375227476060544} a^{9} - \frac{9403934625025675369}{484936125682428181632} a^{8} + \frac{1907259115900824701}{40411343806869015136} a^{7} + \frac{77392226191416090245}{484936125682428181632} a^{6} - \frac{19117019884132833753}{161645375227476060544} a^{5} + \frac{4558442370965786229}{40411343806869015136} a^{4} + \frac{16589055693480241055}{60617015710303522704} a^{3} + \frac{13707631393742480885}{60617015710303522704} a^{2} - \frac{1065740900779901275}{15154253927575880676} a + \frac{24473847695462990}{88106127485906283}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 11728094.3015 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

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 $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-323}) \), \(\Q(\sqrt{17}, \sqrt{-19})\), 4.2.93347.1 x2, 4.0.1773593.1 x2, 8.0.3145632129649.1, 8.2.2814512958107.2 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$