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Properties

Label 16.0.10526401295...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 5^{12}\cdot 89^{4}$
Root discriminant $48.85$
Ramified primes $2, 5, 89$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T210)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![114876, 211296, 266416, 95920, -37408, -35792, 18776, 11584, -3448, -1920, 1264, 224, -220, 16, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 + 16*x^13 - 220*x^12 + 224*x^11 + 1264*x^10 - 1920*x^9 - 3448*x^8 + 11584*x^7 + 18776*x^6 - 35792*x^5 - 37408*x^4 + 95920*x^3 + 266416*x^2 + 211296*x + 114876)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 + 16*x^13 - 220*x^12 + 224*x^11 + 1264*x^10 - 1920*x^9 - 3448*x^8 + 11584*x^7 + 18776*x^6 - 35792*x^5 - 37408*x^4 + 95920*x^3 + 266416*x^2 + 211296*x + 114876, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} + 16 x^{13} - 220 x^{12} + 224 x^{11} + 1264 x^{10} - 1920 x^{9} - 3448 x^{8} + 11584 x^{7} + 18776 x^{6} - 35792 x^{5} - 37408 x^{4} + 95920 x^{3} + 266416 x^{2} + 211296 x + 114876 \)

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:  \(1052640129581056000000000000=2^{36}\cdot 5^{12}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.85$
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:  $2, 5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} + \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{8} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{9} - \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{3204} a^{14} - \frac{41}{1068} a^{13} - \frac{133}{3204} a^{12} - \frac{25}{3204} a^{11} + \frac{43}{3204} a^{10} + \frac{389}{3204} a^{9} - \frac{35}{3204} a^{8} + \frac{173}{534} a^{7} - \frac{80}{801} a^{6} - \frac{27}{178} a^{5} - \frac{451}{1602} a^{4} - \frac{31}{534} a^{3} + \frac{17}{1602} a^{2} - \frac{187}{534} a + \frac{57}{178}$, $\frac{1}{4863604868484575166020007877716} a^{15} - \frac{146489052245289786054888859}{1215901217121143791505001969429} a^{14} - \frac{102863995207802770006041153913}{4863604868484575166020007877716} a^{13} + \frac{59078979882906514904547169459}{1621201622828191722006669292572} a^{12} - \frac{80179212391435973936259842405}{2431802434242287583010003938858} a^{11} + \frac{16073985720167516797869966076}{1215901217121143791505001969429} a^{10} - \frac{250285875981863347719683582263}{4863604868484575166020007877716} a^{9} - \frac{557989699034867475096775852501}{4863604868484575166020007877716} a^{8} + \frac{354532421690641514501969251624}{1215901217121143791505001969429} a^{7} + \frac{159926015781966982144380972379}{2431802434242287583010003938858} a^{6} - \frac{1171894866229944206743700502961}{2431802434242287583010003938858} a^{5} + \frac{775361701193507372896954610467}{2431802434242287583010003938858} a^{4} + \frac{130834420046626836397099454503}{1215901217121143791505001969429} a^{3} + \frac{285423404686714308081153153866}{1215901217121143791505001969429} a^{2} - \frac{3514853120273983799000471195}{62353908570315066231025742022} a - \frac{61480719162831265020943259569}{270200270471365287001111548762}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 549245.028919 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T210):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.178000.1, 4.4.712000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.3645440000.8, 8.8.8111104000000.2, 8.0.91136000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
5Data not computed
89Data not computed