↑

Properties

Label 16.0.15434889281...0944.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 257^{14}$
Root discriminant $1027.50$
Ramified primes $2, 257$
Class number $38163972096$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 4, 8, 24, 24, 32352]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5648874013696, 0, -2243290349568, 0, 164025558208, 0, 2840635392, 0, 2073969697, 0, 30716640, 0, 98688, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 98688*x^12 + 30716640*x^10 + 2073969697*x^8 + 2840635392*x^6 + 164025558208*x^4 - 2243290349568*x^2 + 5648874013696)
 
gp: K = bnfinit(x^16 + 98688*x^12 + 30716640*x^10 + 2073969697*x^8 + 2840635392*x^6 + 164025558208*x^4 - 2243290349568*x^2 + 5648874013696, 1)
 

Normalized defining polynomial

\( x^{16} + 98688 x^{12} + 30716640 x^{10} + 2073969697 x^{8} + 2840635392 x^{6} + 164025558208 x^{4} - 2243290349568 x^{2} + 5648874013696 \)

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:  \(1543488928107008169638324168125095939257028050944=2^{48}\cdot 257^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1027.50$
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:  $2, 257$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4112=2^{4}\cdot 257\)
Dirichlet character group:    $\lbrace$$\chi_{4112}(1,·)$, $\chi_{4112}(835,·)$, $\chi_{4112}(1221,·)$, $\chi_{4112}(1349,·)$, $\chi_{4112}(513,·)$, $\chi_{4112}(3851,·)$, $\chi_{4112}(1803,·)$, $\chi_{4112}(2317,·)$, $\chi_{4112}(707,·)$, $\chi_{4112}(2055,·)$, $\chi_{4112}(2329,·)$, $\chi_{4112}(3871,·)$, $\chi_{4112}(1543,·)$, $\chi_{4112}(2297,·)$, $\chi_{4112}(253,·)$, $\chi_{4112}(3839,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{257} a^{8}$, $\frac{1}{8738} a^{9} + \frac{5}{17} a^{7} - \frac{2}{17} a^{5} - \frac{3}{17} a^{3} - \frac{9}{34} a$, $\frac{1}{17476} a^{10} + \frac{5}{4369} a^{8} - \frac{1}{17} a^{6} + \frac{7}{17} a^{4} + \frac{25}{68} a^{2}$, $\frac{1}{594184} a^{11} - \frac{4}{74273} a^{9} + \frac{45}{289} a^{7} - \frac{115}{289} a^{5} + \frac{745}{2312} a^{3} + \frac{84}{289} a$, $\frac{1}{161618048} a^{12} + \frac{287}{10101128} a^{10} + \frac{2241}{5050564} a^{8} - \frac{7037}{19652} a^{6} + \frac{90913}{628864} a^{4} - \frac{10073}{39304} a^{2} - \frac{1}{4}$, $\frac{1}{323236096} a^{13} + \frac{15}{20202256} a^{11} - \frac{343}{10101128} a^{9} + \frac{12207}{39304} a^{7} - \frac{48351}{1257728} a^{5} - \frac{11569}{78608} a^{3} - \frac{837}{2312} a$, $\frac{1}{555309384020555938339855979446784} a^{14} - \frac{34623411764108417142357}{138827346005138984584963994861696} a^{12} + \frac{490945256252411834960086419}{17353418250642373073120499357712} a^{10} - \frac{883283282560297211242051757}{17353418250642373073120499357712} a^{8} + \frac{290822388190963902045803653537}{2160736902803719604435237274112} a^{6} - \frac{91514553459743211713806639093}{540184225700929901108809318528} a^{4} + \frac{18953376568059375079286579197}{67523028212616237638601164816} a^{2} - \frac{66434509945802226103575}{202113924080818709182724}$, $\frac{1}{2221237536082223753359423917787136} a^{15} + \frac{206090191394842774221055}{138827346005138984584963994861696} a^{13} + \frac{49427389097863364750425841}{69413673002569492292481997430848} a^{11} - \frac{2996384358825256815747431177}{69413673002569492292481997430848} a^{9} - \frac{191308250868384279587033485023}{8642947611214878417740949096448} a^{7} + \frac{143372810000874932004186665759}{540184225700929901108809318528} a^{5} + \frac{83890862406871739380667747243}{270092112850464950554404659264} a^{3} - \frac{2137908264833602031296902}{14602731014839151738451809} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{8}\times C_{24}\times C_{24}\times C_{32352}$, which has order $38163972096$ (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:  \( 56884691.566921 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{257}) \), \(\Q(\sqrt{-514}) \), \(\Q(\sqrt{-2}, \sqrt{257})\), 4.4.1086373952.2, 4.0.271593488.2, 8.0.18883333817345572864.16, 8.0.310593074627699982467072.6, 8.8.310593074627699982467072.2

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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$2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
257Data not computed