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Properties

Label 16.8.37503454031...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{6}\cdot 7^{6}\cdot 41^{6}$
Root discriminant $61.08$
Ramified primes $2, 5, 7, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1189

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![165953, -1271952, -2728132, -2446848, -692715, 226144, 125608, 64448, 29902, 576, -302, -384, -288, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 288*x^12 - 384*x^11 - 302*x^10 + 576*x^9 + 29902*x^8 + 64448*x^7 + 125608*x^6 + 226144*x^5 - 692715*x^4 - 2446848*x^3 - 2728132*x^2 - 1271952*x + 165953)
 
gp: K = bnfinit(x^16 - 4*x^14 - 288*x^12 - 384*x^11 - 302*x^10 + 576*x^9 + 29902*x^8 + 64448*x^7 + 125608*x^6 + 226144*x^5 - 692715*x^4 - 2446848*x^3 - 2728132*x^2 - 1271952*x + 165953, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 288 x^{12} - 384 x^{11} - 302 x^{10} + 576 x^{9} + 29902 x^{8} + 64448 x^{7} + 125608 x^{6} + 226144 x^{5} - 692715 x^{4} - 2446848 x^{3} - 2728132 x^{2} - 1271952 x + 165953 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(37503454031498105061376000000=2^{32}\cdot 5^{6}\cdot 7^{6}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.08$
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:  $2, 5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{287} a^{12} - \frac{48}{287} a^{11} + \frac{1}{287} a^{10} + \frac{72}{287} a^{9} - \frac{12}{41} a^{8} - \frac{24}{287} a^{7} - \frac{124}{287} a^{6} - \frac{12}{287} a^{5} - \frac{3}{7} a^{4} - \frac{102}{287} a^{3} + \frac{33}{287} a^{2} - \frac{57}{287} a - \frac{19}{287}$, $\frac{1}{287} a^{13} - \frac{1}{41} a^{11} + \frac{120}{287} a^{10} - \frac{72}{287} a^{9} - \frac{38}{287} a^{8} - \frac{128}{287} a^{7} + \frac{9}{41} a^{6} - \frac{125}{287} a^{5} + \frac{3}{41} a^{4} + \frac{16}{287} a^{3} + \frac{92}{287} a^{2} + \frac{115}{287} a - \frac{51}{287}$, $\frac{1}{82369} a^{14} - \frac{24}{82369} a^{13} + \frac{141}{82369} a^{12} + \frac{6960}{82369} a^{11} - \frac{31217}{82369} a^{10} - \frac{4874}{82369} a^{9} + \frac{1780}{11767} a^{8} - \frac{1852}{82369} a^{7} - \frac{6213}{82369} a^{6} - \frac{6791}{82369} a^{5} - \frac{31033}{82369} a^{4} - \frac{29738}{82369} a^{3} - \frac{29927}{82369} a^{2} - \frac{15265}{82369} a - \frac{16512}{82369}$, $\frac{1}{1118843002966151765379429919334410769} a^{15} - \frac{160996112345750285684127566080}{27288853730881750375108046813034409} a^{14} - \frac{247591349591492209199968596781980}{159834714709450252197061417047772967} a^{13} + \frac{53825151776906884957993285162722}{159834714709450252197061417047772967} a^{12} - \frac{179809090674910762419954056788159564}{1118843002966151765379429919334410769} a^{11} + \frac{320557041660057434847402283809310024}{1118843002966151765379429919334410769} a^{10} - \frac{434750277990446045377040555242053239}{1118843002966151765379429919334410769} a^{9} - \frac{324118808840555307511912461806492893}{1118843002966151765379429919334410769} a^{8} - \frac{198991569043437020020058940766353999}{1118843002966151765379429919334410769} a^{7} - \frac{45879263988316462100441678822327682}{159834714709450252197061417047772967} a^{6} + \frac{34472161795705687654676816425074847}{159834714709450252197061417047772967} a^{5} + \frac{183508677750227231448042056073775554}{1118843002966151765379429919334410769} a^{4} - \frac{428208960197751825372452897707048078}{1118843002966151765379429919334410769} a^{3} - \frac{310079172923261958260195012317429499}{1118843002966151765379429919334410769} a^{2} + \frac{8295980791352908996197141211571482}{1118843002966151765379429919334410769} a + \frac{425830815863352126869051830389633459}{1118843002966151765379429919334410769}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 102717106.23 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1189:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1189 are not computed
Character table for t16n1189 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.8434585600.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.8.6.3$x^{8} + 25 x^{4} + 200$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$7$7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$