↑

Properties

Label 16.0.16697436405...9977.3
Degree $16$
Signature $[0, 8]$
Discriminant $37^{4}\cdot 73^{15}$
Root discriminant $137.69$
Ramified primes $37, 73$
Class number $178$ (GRH)
Class group $[178]$ (GRH)
Galois group 16T841

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![102711296, -67074048, -4564992, 4748416, 25364608, -28593728, 14199152, -3576936, 301680, 59080, -8753, -1993, 233, 80, 7, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 7*x^14 + 80*x^13 + 233*x^12 - 1993*x^11 - 8753*x^10 + 59080*x^9 + 301680*x^8 - 3576936*x^7 + 14199152*x^6 - 28593728*x^5 + 25364608*x^4 + 4748416*x^3 - 4564992*x^2 - 67074048*x + 102711296)
 
gp: K = bnfinit(x^16 - 7*x^15 + 7*x^14 + 80*x^13 + 233*x^12 - 1993*x^11 - 8753*x^10 + 59080*x^9 + 301680*x^8 - 3576936*x^7 + 14199152*x^6 - 28593728*x^5 + 25364608*x^4 + 4748416*x^3 - 4564992*x^2 - 67074048*x + 102711296, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 7 x^{14} + 80 x^{13} + 233 x^{12} - 1993 x^{11} - 8753 x^{10} + 59080 x^{9} + 301680 x^{8} - 3576936 x^{7} + 14199152 x^{6} - 28593728 x^{5} + 25364608 x^{4} + 4748416 x^{3} - 4564992 x^{2} - 67074048 x + 102711296 \)

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:  \(16697436405715421965395561284159977=37^{4}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $137.69$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{9} - \frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{1}{4} a^{5} - \frac{1}{32} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{10} + \frac{3}{64} a^{8} - \frac{3}{64} a^{7} - \frac{1}{8} a^{6} + \frac{3}{64} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{9} - \frac{3}{64} a^{8} - \frac{1}{16} a^{7} - \frac{1}{64} a^{6} + \frac{3}{16} a^{5} - \frac{3}{16} a^{4} - \frac{3}{8} a^{3}$, $\frac{1}{512} a^{12} - \frac{1}{512} a^{11} - \frac{3}{512} a^{10} - \frac{3}{256} a^{9} + \frac{9}{512} a^{8} + \frac{29}{512} a^{7} - \frac{39}{512} a^{6} - \frac{55}{256} a^{5} + \frac{5}{32} a^{4} + \frac{31}{64} a^{3} - \frac{5}{32} a^{2} - \frac{3}{8} a$, $\frac{1}{2048} a^{13} + \frac{1}{2048} a^{12} - \frac{5}{2048} a^{11} + \frac{1}{512} a^{10} + \frac{29}{2048} a^{9} + \frac{31}{2048} a^{8} - \frac{125}{2048} a^{7} - \frac{39}{512} a^{6} - \frac{39}{512} a^{5} + \frac{63}{256} a^{4} - \frac{23}{64} a^{3} + \frac{17}{64} a^{2} - \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{606208} a^{14} - \frac{131}{606208} a^{13} + \frac{567}{606208} a^{12} + \frac{147}{75776} a^{11} + \frac{3789}{606208} a^{10} - \frac{49}{16384} a^{9} + \frac{32007}{606208} a^{8} + \frac{3987}{75776} a^{7} + \frac{8501}{151552} a^{6} + \frac{3325}{75776} a^{5} - \frac{1395}{9472} a^{4} - \frac{147}{18944} a^{3} - \frac{47}{2368} a^{2} - \frac{521}{1184} a - \frac{53}{148}$, $\frac{1}{24253687193820298572564906551500341248} a^{15} - \frac{462034704814364934521325776537}{24253687193820298572564906551500341248} a^{14} + \frac{5400382191659450764325027384196169}{24253687193820298572564906551500341248} a^{13} + \frac{1308805411156982849388681683157719}{12126843596910149286282453275750170624} a^{12} - \frac{1974078433097516830162290190815767}{655505059292440501961213690581090304} a^{11} + \frac{73714012048672400308841300820886925}{24253687193820298572564906551500341248} a^{10} + \frac{118799847375770285189437881995822181}{24253687193820298572564906551500341248} a^{9} - \frac{384870820342457849996298291593485545}{12126843596910149286282453275750170624} a^{8} + \frac{72716788833727858667176020450397645}{6063421798455074643141226637875085312} a^{7} + \frac{139430705374100040861741989045945631}{1515855449613768660785306659468771328} a^{6} - \frac{2508367924815630668095019131725131}{40969066205777531372575855661318144} a^{5} - \frac{133869725217942596911017216384261283}{757927724806884330392653329734385664} a^{4} + \frac{41022170137553942943835568403153885}{378963862403442165196326664867192832} a^{3} - \frac{12700015161449173267924453092231351}{47370482800430270649540833108399104} a^{2} + \frac{8188951462670157213782274162551605}{23685241400215135324770416554199552} a + \frac{1392543431382914154919326370085685}{2960655175026891915596302069274944}$

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

Class group and class number

$C_{178}$, which has order $178$ (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:  \( 724794041959 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.0.11047398519097.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ $16$ $16$ $16$

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
37Data not computed
73Data not computed