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Properties

Label 16.0.33738487444...7936.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{56}\cdot 193^{2}\cdot 257^{10}$
Root discriminant $700.66$
Ramified primes $2, 193, 257$
Class number $4480628736$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 4, 17502456]$ (GRH)
Galois group 16T1435

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41236000985452036, 0, 4937542579833672, 0, 199890202805044, 0, 3727038571256, 0, 37699023596, 0, 220686820, 0, 746298, 0, 1348, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1348*x^14 + 746298*x^12 + 220686820*x^10 + 37699023596*x^8 + 3727038571256*x^6 + 199890202805044*x^4 + 4937542579833672*x^2 + 41236000985452036)
 
gp: K = bnfinit(x^16 + 1348*x^14 + 746298*x^12 + 220686820*x^10 + 37699023596*x^8 + 3727038571256*x^6 + 199890202805044*x^4 + 4937542579833672*x^2 + 41236000985452036, 1)
 

Normalized defining polynomial

\( x^{16} + 1348 x^{14} + 746298 x^{12} + 220686820 x^{10} + 37699023596 x^{8} + 3727038571256 x^{6} + 199890202805044 x^{4} + 4937542579833672 x^{2} + 41236000985452036 \)

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:  \(3373848744484085385907843975364516141319847936=2^{56}\cdot 193^{2}\cdot 257^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $700.66$
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:  $2, 193, 257$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9252} a^{12} - \frac{59}{771} a^{10} - \frac{136}{2313} a^{8} + \frac{230}{2313} a^{6} - \frac{1315}{4626} a^{4} + \frac{1}{9} a^{2} - \frac{1}{9}$, $\frac{1}{9252} a^{13} - \frac{59}{771} a^{11} - \frac{136}{2313} a^{9} + \frac{230}{2313} a^{7} - \frac{1315}{4626} a^{5} + \frac{1}{9} a^{3} - \frac{1}{9} a$, $\frac{1}{2233461418368523265028745100396616219540} a^{14} + \frac{25032859169230880290595711389466339}{558365354592130816257186275099154054885} a^{12} + \frac{10939268766162820622871527146378521119}{558365354592130816257186275099154054885} a^{10} + \frac{7934230081255860805928038819918822781}{124081189909362403612708061133145345530} a^{8} - \frac{8623684195819180427447882146354339795}{223346141836852326502874510039661621954} a^{6} - \frac{187593017780407361738093442479692580366}{558365354592130816257186275099154054885} a^{4} - \frac{685024247249374123715518702866846161}{2172627838879886444580491342798264805} a^{2} - \frac{1888393033885990465043561785548953}{11257139061553815775028452553358885}$, $\frac{1}{4571895523400367123513841220511873401398380} a^{15} + \frac{23948971946017212650966596193559845533}{2285947761700183561756920610255936700699190} a^{13} + \frac{76236138577307725471099689961661633694674}{1142973880850091780878460305127968350349595} a^{11} - \frac{18932069656900758657960315269320776740629}{253994195744464840195213401139548522299910} a^{9} - \frac{11766595864501913966971122963858190362481}{457189552340036712351384122051187340139838} a^{7} - \frac{5194092920130719414799661217318412939997}{49694516558699642646889578483824710884765} a^{5} - \frac{4469501420634958131260809209321099652}{193363877660309893567663729509045567645} a^{3} + \frac{4174510198802579662070512335510597382}{23043363659000660891483242376725637595} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{17502456}$, which has order $4480628736$ (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:  \( 9417883.23016 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1435:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 56 conjugacy class representatives for t16n1435 are not computed
Character table for t16n1435 is not computed

Intermediate fields

\(\Q(\sqrt{514}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{257}) \), \(\Q(\sqrt{2}, \sqrt{257})\), 8.8.2287190881599488.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.31.10$x^{8} + 24 x^{6} + 12 x^{4} + 2$$8$$1$$31$$C_8:C_2$$[3, 4, 5]^{2}$
2.8.25.2$x^{8} + 10 x^{4} + 20 x^{2} + 2$$8$$1$$25$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
193Data not computed
257Data not computed