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Properties

Label 16.0.28621836185...801.23
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 53^{12}$
Root discriminant $164.45$
Ramified primes $17, 53$
Class number $313600$ (GRH)
Class group $[2, 280, 560]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15061337953, 33661076896, 32555640354, 18036657805, 6943110997, 1894750586, 412915231, 72959856, 14494211, 2076975, 442958, 19563, 13024, -159, 183, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 183*x^14 - 159*x^13 + 13024*x^12 + 19563*x^11 + 442958*x^10 + 2076975*x^9 + 14494211*x^8 + 72959856*x^7 + 412915231*x^6 + 1894750586*x^5 + 6943110997*x^4 + 18036657805*x^3 + 32555640354*x^2 + 33661076896*x + 15061337953)
 
gp: K = bnfinit(x^16 - 3*x^15 + 183*x^14 - 159*x^13 + 13024*x^12 + 19563*x^11 + 442958*x^10 + 2076975*x^9 + 14494211*x^8 + 72959856*x^7 + 412915231*x^6 + 1894750586*x^5 + 6943110997*x^4 + 18036657805*x^3 + 32555640354*x^2 + 33661076896*x + 15061337953, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 183 x^{14} - 159 x^{13} + 13024 x^{12} + 19563 x^{11} + 442958 x^{10} + 2076975 x^{9} + 14494211 x^{8} + 72959856 x^{7} + 412915231 x^{6} + 1894750586 x^{5} + 6943110997 x^{4} + 18036657805 x^{3} + 32555640354 x^{2} + 33661076896 x + 15061337953 \)

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:  \(286218361857094751614277009933470801=17^{12}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $164.45$
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:  $17, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{15693880299761037451719874365336987467250428140233511537701040977690730898422} a^{15} - \frac{2938007885434619602625447851728186153726117600365049832224288133595573981069}{15693880299761037451719874365336987467250428140233511537701040977690730898422} a^{14} + \frac{207960977498476882404086463746739458162607524201235728980058581667669800361}{7846940149880518725859937182668493733625214070116755768850520488845365449211} a^{13} + \frac{1936135237797839013420354320727247706038802738019455404667997349362233394046}{7846940149880518725859937182668493733625214070116755768850520488845365449211} a^{12} - \frac{323863080107502702525273940910650576985333570083996510348290722735709464261}{7846940149880518725859937182668493733625214070116755768850520488845365449211} a^{11} - \frac{3067509663657547490789118853213463501359688254242427046350010559086885095791}{15693880299761037451719874365336987467250428140233511537701040977690730898422} a^{10} - \frac{457442389383020695056502230728589438047407783954990573751969408117993316105}{15693880299761037451719874365336987467250428140233511537701040977690730898422} a^{9} + \frac{43734496071919760399934525978399512690248918780104904812215901201730907338}{603610780760039901989225937128345671817324159239750443757732345295797342247} a^{8} - \frac{644791336187082811976360450741708213856301560438461302195830606067990836948}{7846940149880518725859937182668493733625214070116755768850520488845365449211} a^{7} + \frac{1222445332869185147309465151767510646973697704580049352918082995526804863524}{7846940149880518725859937182668493733625214070116755768850520488845365449211} a^{6} + \frac{2799241817933174715903264587797467677705877900866300958921899347216196370865}{15693880299761037451719874365336987467250428140233511537701040977690730898422} a^{5} - \frac{6275094920357775844564723392734934371003175981116198354225049862093995055951}{15693880299761037451719874365336987467250428140233511537701040977690730898422} a^{4} + \frac{9470301059106265632350912616420777099374235891109123602248316158473965712}{603610780760039901989225937128345671817324159239750443757732345295797342247} a^{3} - \frac{181159798490963590554023619491931731463408487303638852419912282201302297778}{7846940149880518725859937182668493733625214070116755768850520488845365449211} a^{2} - \frac{3251611591614754881920282071814689005142314171172245762818527074091892331761}{7846940149880518725859937182668493733625214070116755768850520488845365449211} a - \frac{33950151294014620322900675149240602115148374622594088557235297482465064247}{166956173401713164379998663461031781566493916385462888698947244443518413813}$

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

Class group and class number

$C_{2}\times C_{280}\times C_{560}$, which has order $313600$ (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:  \( 452391.453967 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{901}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{17}, \sqrt{53})\), 4.4.47753.1 x2, 4.4.15317.1 x2, 4.0.731432701.2, 4.0.731432701.1, 8.8.659020863601.1, 8.0.534993796092155401.6, 8.0.534993796092155401.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ 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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$