↑

Properties

Label 16.0.10614152867...6401.2
Degree $16$
Signature $[0, 8]$
Discriminant $23^{4}\cdot 41^{14}$
Root discriminant $56.44$
Ramified primes $23, 41$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![385867, 960577, 333897, -115408, 423774, -182893, 67069, -13606, 10279, -3083, 967, -104, 102, -16, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 - 16*x^13 + 102*x^12 - 104*x^11 + 967*x^10 - 3083*x^9 + 10279*x^8 - 13606*x^7 + 67069*x^6 - 182893*x^5 + 423774*x^4 - 115408*x^3 + 333897*x^2 + 960577*x + 385867)
 
gp: K = bnfinit(x^16 - 2*x^15 + 6*x^14 - 16*x^13 + 102*x^12 - 104*x^11 + 967*x^10 - 3083*x^9 + 10279*x^8 - 13606*x^7 + 67069*x^6 - 182893*x^5 + 423774*x^4 - 115408*x^3 + 333897*x^2 + 960577*x + 385867, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 6 x^{14} - 16 x^{13} + 102 x^{12} - 104 x^{11} + 967 x^{10} - 3083 x^{9} + 10279 x^{8} - 13606 x^{7} + 67069 x^{6} - 182893 x^{5} + 423774 x^{4} - 115408 x^{3} + 333897 x^{2} + 960577 x + 385867 \)

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:  \(10614152867452364890755536401=23^{4}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.44$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{472} a^{12} - \frac{15}{118} a^{11} + \frac{89}{472} a^{10} + \frac{105}{236} a^{9} - \frac{75}{236} a^{8} - \frac{93}{236} a^{7} - \frac{77}{236} a^{6} + \frac{69}{472} a^{5} + \frac{113}{472} a^{4} - \frac{67}{472} a^{3} + \frac{39}{118} a^{2} + \frac{99}{472} a + \frac{201}{472}$, $\frac{1}{472} a^{13} + \frac{29}{472} a^{11} - \frac{57}{236} a^{10} + \frac{89}{236} a^{9} + \frac{9}{236} a^{8} - \frac{111}{236} a^{7} - \frac{203}{472} a^{6} - \frac{231}{472} a^{5} + \frac{105}{472} a^{4} + \frac{37}{118} a^{3} - \frac{217}{472} a^{2} - \frac{231}{472} a + \frac{3}{59}$, $\frac{1}{78352} a^{14} - \frac{1}{78352} a^{13} + \frac{37}{39176} a^{12} + \frac{19577}{78352} a^{11} + \frac{4533}{78352} a^{10} + \frac{2167}{39176} a^{9} - \frac{17419}{39176} a^{8} - \frac{35019}{78352} a^{7} + \frac{10799}{39176} a^{6} - \frac{12607}{78352} a^{5} - \frac{1125}{4897} a^{4} - \frac{1229}{4897} a^{3} + \frac{13651}{39176} a^{2} - \frac{4843}{39176} a - \frac{369}{944}$, $\frac{1}{98653875322043463859891595792298512} a^{15} + \frac{608698129035429465327664421879}{98653875322043463859891595792298512} a^{14} - \frac{400553639677485011877191115538}{6165867207627716491243224737018657} a^{13} + \frac{33907968505134479894110188590817}{98653875322043463859891595792298512} a^{12} + \frac{16542714095019518114571011052994595}{98653875322043463859891595792298512} a^{11} + \frac{6376648851524665519758036727846465}{49326937661021731929945797896149256} a^{10} + \frac{1528943426044154049411326237036355}{49326937661021731929945797896149256} a^{9} + \frac{11579722686663075478148080153887921}{98653875322043463859891595792298512} a^{8} - \frac{2696412835364773691211358963504487}{49326937661021731929945797896149256} a^{7} - \frac{21773374446884258560190094979207065}{98653875322043463859891595792298512} a^{6} + \frac{6662580014706679161923273978977631}{49326937661021731929945797896149256} a^{5} - \frac{21155967412486395663499912400302581}{49326937661021731929945797896149256} a^{4} - \frac{22898504378253470872695936208929069}{49326937661021731929945797896149256} a^{3} - \frac{4449182664880744276156037179848893}{24663468830510865964972898948074628} a^{2} - \frac{19905001898798296058064670727078237}{98653875322043463859891595792298512} a - \frac{30573903102727624115481044016044}{74287556718406222786062948638779}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 3403166.59806 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.2.38663.1, 4.4.68921.1, 4.2.1585183.1, 8.0.194754273881.1, 8.4.103025010883049.3, 8.4.2512805143489.1

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 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$