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Properties

Label 16.0.16697436405...9977.1
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

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19765336, -43706312, 36407032, -12386043, 2146605, -2215390, 1686286, -415744, 66258, -55432, 18306, -2378, 786, -136, 14, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 136*x^13 + 786*x^12 - 2378*x^11 + 18306*x^10 - 55432*x^9 + 66258*x^8 - 415744*x^7 + 1686286*x^6 - 2215390*x^5 + 2146605*x^4 - 12386043*x^3 + 36407032*x^2 - 43706312*x + 19765336)
 
gp: K = bnfinit(x^16 - 5*x^15 + 14*x^14 - 136*x^13 + 786*x^12 - 2378*x^11 + 18306*x^10 - 55432*x^9 + 66258*x^8 - 415744*x^7 + 1686286*x^6 - 2215390*x^5 + 2146605*x^4 - 12386043*x^3 + 36407032*x^2 - 43706312*x + 19765336, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 14 x^{14} - 136 x^{13} + 786 x^{12} - 2378 x^{11} + 18306 x^{10} - 55432 x^{9} + 66258 x^{8} - 415744 x^{7} + 1686286 x^{6} - 2215390 x^{5} + 2146605 x^{4} - 12386043 x^{3} + 36407032 x^{2} - 43706312 x + 19765336 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{12} - \frac{3}{32} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{7}{32} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{13}{32} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{32} a^{13} - \frac{3}{32} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{7}{32} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{3}{32} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{1}{64} a^{12} + \frac{1}{64} a^{11} - \frac{5}{64} a^{10} + \frac{7}{64} a^{9} - \frac{1}{64} a^{8} - \frac{15}{64} a^{7} - \frac{15}{64} a^{6} + \frac{15}{64} a^{5} + \frac{13}{64} a^{4} - \frac{31}{64} a^{3} + \frac{1}{4} a^{2} + \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{4449751549381749633535818296986921119578699264} a^{15} - \frac{7356258733723385136219889752123493821970877}{2224875774690874816767909148493460559789349632} a^{14} - \frac{749886005874653470659853478487274038706019}{69527367959089838023997160890420642493417176} a^{13} + \frac{7940982763771614482410890627033792137659871}{556218943672718704191977287123365139947337408} a^{12} - \frac{27635541987780139967189922683527984763930587}{2224875774690874816767909148493460559789349632} a^{11} - \frac{108417732391088946046634098258865019544340059}{1112437887345437408383954574246730279894674816} a^{10} + \frac{28956450263688971730298362305021770585591599}{2224875774690874816767909148493460559789349632} a^{9} - \frac{219212616666387049736739472483178324645866487}{2224875774690874816767909148493460559789349632} a^{8} + \frac{45792495186176131565865551971260960139962141}{556218943672718704191977287123365139947337408} a^{7} - \frac{22241746644919535118381835588830759852143785}{556218943672718704191977287123365139947337408} a^{6} + \frac{243530354263069375940676502993297144170523923}{2224875774690874816767909148493460559789349632} a^{5} - \frac{153053223559793799067014922580730004283336235}{1112437887345437408383954574246730279894674816} a^{4} + \frac{780282001794963188764599182297880798222892233}{4449751549381749633535818296986921119578699264} a^{3} + \frac{35802869951033314543751482333553248611230299}{556218943672718704191977287123365139947337408} a^{2} - \frac{19383349567183821703310294338841257252513713}{69527367959089838023997160890420642493417176} a + \frac{227889878554967987720920312634895491454752335}{556218943672718704191977287123365139947337408}$

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:  \( 8268925124.35 \) (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ 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