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Properties

Label 16.0.10407187972...0944.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 17^{6}\cdot 89^{4}$
Root discriminant $42.27$
Ramified primes $2, 17, 89$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8602768, -10335520, 21996912, -15726656, 14169936, -6791264, 3829968, -1285200, 533146, -129312, 44064, -7772, 2272, -256, 70, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 70*x^14 - 256*x^13 + 2272*x^12 - 7772*x^11 + 44064*x^10 - 129312*x^9 + 533146*x^8 - 1285200*x^7 + 3829968*x^6 - 6791264*x^5 + 14169936*x^4 - 15726656*x^3 + 21996912*x^2 - 10335520*x + 8602768)
 
gp: K = bnfinit(x^16 - 4*x^15 + 70*x^14 - 256*x^13 + 2272*x^12 - 7772*x^11 + 44064*x^10 - 129312*x^9 + 533146*x^8 - 1285200*x^7 + 3829968*x^6 - 6791264*x^5 + 14169936*x^4 - 15726656*x^3 + 21996912*x^2 - 10335520*x + 8602768, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 70 x^{14} - 256 x^{13} + 2272 x^{12} - 7772 x^{11} + 44064 x^{10} - 129312 x^{9} + 533146 x^{8} - 1285200 x^{7} + 3829968 x^{6} - 6791264 x^{5} + 14169936 x^{4} - 15726656 x^{3} + 21996912 x^{2} - 10335520 x + 8602768 \)

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:  \(104071879720680162479570944=2^{36}\cdot 17^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.27$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} + \frac{1}{12} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{14} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{4} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{49666897192366736178082687686710799530283222744} a^{15} - \frac{132301834365925190717933312425523561694066899}{24833448596183368089041343843355399765141611372} a^{14} - \frac{273085737027659252062282962667681675322075291}{24833448596183368089041343843355399765141611372} a^{13} + \frac{54104443041422248012732098233684223715510417}{4138908099363894681506890640559233294190268562} a^{12} + \frac{647164118935689337542969910315863517030453448}{6208362149045842022260335960838849941285402843} a^{11} + \frac{2493040482850965429278669540060410106134584005}{12416724298091684044520671921677699882570805686} a^{10} + \frac{188553403054886888048653104945589412277795552}{2069454049681947340753445320279616647095134281} a^{9} + \frac{299627664259246105299323726833475498547801609}{4138908099363894681506890640559233294190268562} a^{8} - \frac{594162901613431638858435741899812567767904297}{8277816198727789363013781281118466588380537124} a^{7} - \frac{757932647791874122942651283276835789381418951}{4138908099363894681506890640559233294190268562} a^{6} - \frac{3053879480385310950667867070231163123972483776}{6208362149045842022260335960838849941285402843} a^{5} - \frac{2780882263583866595908974376761777820663750531}{6208362149045842022260335960838849941285402843} a^{4} + \frac{1107697513351312152149110296017784259829246387}{6208362149045842022260335960838849941285402843} a^{3} - \frac{926712951136802512101825580643205215910687099}{2069454049681947340753445320279616647095134281} a^{2} + \frac{1452192156240222836210606947262141416998537820}{6208362149045842022260335960838849941285402843} a - \frac{2934462805674205041345914441064107281264327027}{6208362149045842022260335960838849941285402843}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( \frac{12841097429160668758536680706856615}{92235189713967133804317492514519174363224} a^{15} + \frac{31772668368610170389379045581717345}{92235189713967133804317492514519174363224} a^{14} + \frac{288492199299987410765574916884576247}{46117594856983566902158746257259587181612} a^{13} + \frac{84904251631615797662580440614357284}{3843132904748630575179895521438298931801} a^{12} + \frac{1261383717396656650493421195185030714}{11529398714245891725539686564314896795403} a^{11} + \frac{14453589235738275502426022403979243241}{23058797428491783451079373128629793590806} a^{10} + \frac{1182505600322427988640899094235865492}{3843132904748630575179895521438298931801} a^{9} + \frac{45537012135093497093619669179102323762}{3843132904748630575179895521438298931801} a^{8} - \frac{178812835671606294532008839090648380675}{15372531618994522300719582085753195727204} a^{7} + \frac{2071498110138867775650490453100649207447}{15372531618994522300719582085753195727204} a^{6} - \frac{2248249646584571026960347142857146770625}{11529398714245891725539686564314896795403} a^{5} + \frac{18667803696391786255644897076837439102653}{23058797428491783451079373128629793590806} a^{4} - \frac{8132386428383308863719553294672056610949}{11529398714245891725539686564314896795403} a^{3} + \frac{5262063439787373423467895280685476937156}{3843132904748630575179895521438298931801} a^{2} + \frac{3496956813804015253453475283179987285443}{11529398714245891725539686564314896795403} a - \frac{3362444870180339476323621859838346130183}{11529398714245891725539686564314896795403} \) (order $8$)
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:  \( 2861210.36565 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.4.4352.1, 4.0.1088.2, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.18.53$x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.18.53$x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$