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Properties

Label 16.0.24206677051...5849.4
Degree $16$
Signature $[0, 8]$
Discriminant $7^{12}\cdot 53^{10}$
Root discriminant $51.46$
Ramified primes $7, 53$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![220367, -192542, 99876, 6594, 17424, 401, -10699, -9028, 3180, 2522, -287, -479, 162, 44, -6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 6*x^14 + 44*x^13 + 162*x^12 - 479*x^11 - 287*x^10 + 2522*x^9 + 3180*x^8 - 9028*x^7 - 10699*x^6 + 401*x^5 + 17424*x^4 + 6594*x^3 + 99876*x^2 - 192542*x + 220367)
 
gp: K = bnfinit(x^16 - 6*x^15 - 6*x^14 + 44*x^13 + 162*x^12 - 479*x^11 - 287*x^10 + 2522*x^9 + 3180*x^8 - 9028*x^7 - 10699*x^6 + 401*x^5 + 17424*x^4 + 6594*x^3 + 99876*x^2 - 192542*x + 220367, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 6 x^{14} + 44 x^{13} + 162 x^{12} - 479 x^{11} - 287 x^{10} + 2522 x^{9} + 3180 x^{8} - 9028 x^{7} - 10699 x^{6} + 401 x^{5} + 17424 x^{4} + 6594 x^{3} + 99876 x^{2} - 192542 x + 220367 \)

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:  \(2420667705185442556922185849=7^{12}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.46$
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:  $7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4}$, $\frac{1}{14} a^{11} - \frac{1}{14} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{3}{14} a^{6} + \frac{3}{14} a^{5} + \frac{1}{7} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{154} a^{12} - \frac{2}{77} a^{11} - \frac{5}{154} a^{10} + \frac{13}{77} a^{9} + \frac{2}{77} a^{8} + \frac{3}{14} a^{7} - \frac{3}{77} a^{6} + \frac{23}{154} a^{5} - \frac{16}{77} a^{4} + \frac{1}{22} a^{2} - \frac{2}{11} a - \frac{5}{22}$, $\frac{1}{308} a^{13} - \frac{5}{154} a^{11} + \frac{17}{308} a^{10} - \frac{17}{77} a^{9} - \frac{17}{308} a^{8} - \frac{18}{77} a^{7} - \frac{39}{154} a^{6} + \frac{115}{308} a^{5} + \frac{5}{22} a^{4} + \frac{1}{44} a^{3} - \frac{1}{2} a^{2} - \frac{5}{22} a - \frac{9}{44}$, $\frac{1}{303688} a^{14} - \frac{171}{303688} a^{13} - \frac{201}{75922} a^{12} - \frac{1091}{43384} a^{11} + \frac{3767}{303688} a^{10} - \frac{114897}{303688} a^{9} + \frac{11573}{27608} a^{8} + \frac{30003}{75922} a^{7} + \frac{60897}{303688} a^{6} - \frac{2949}{17864} a^{5} - \frac{150455}{303688} a^{4} + \frac{18727}{43384} a^{3} - \frac{7915}{21692} a^{2} - \frac{1019}{43384} a - \frac{5975}{43384}$, $\frac{1}{2798140210063693507537801625272} a^{15} - \frac{19169886022783552181329}{699535052515923376884450406318} a^{14} - \frac{92450702777979932280257569}{57104902246197826684444931128} a^{13} + \frac{7891285976551940359140499163}{2798140210063693507537801625272} a^{12} - \frac{18823592870995652168955051831}{699535052515923376884450406318} a^{11} + \frac{36827938184694051830961}{16975297932876880702867102} a^{10} - \frac{14739390653446545912588454396}{49966789465423098348889314737} a^{9} - \frac{935129738566185290082022940275}{2798140210063693507537801625272} a^{8} + \frac{88719691456896420504262045547}{254376382733063046139800147752} a^{7} + \frac{7295445473289827145420412561}{28552451123098913342222465564} a^{6} + \frac{48643347379550127713482017447}{127188191366531523069900073876} a^{5} + \frac{54631284440555428030751522304}{349767526257961688442225203159} a^{4} - \frac{55009695218400680016453763125}{399734315723384786791114517896} a^{3} + \frac{46153119664097023953030627639}{399734315723384786791114517896} a^{2} - \frac{2091469830685007611739778209}{9084870811895108790707148134} a + \frac{129181198706923558555064252547}{399734315723384786791114517896}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 13170551.5373 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{-371}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{-7}) \), 4.0.2597.2 x2, \(\Q(\sqrt{-7}, \sqrt{53})\), 4.2.19663.1 x2, 8.0.928307199169.1 x2, 8.0.49200281555957.1 x2, 8.0.18945044881.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$7$7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
53Data not computed