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Properties

Label 16.0.50322387234...000.61
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{8}\cdot 17^{8}$
Root discriminant $127.75$
Ramified primes $2, 3, 5, 17$
Class number $4792320$ (GRH)
Class group $[2, 4, 4, 24, 6240]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104415192286, -747884040, 28731561716, -36903144, 3610023514, 2718048, 272216812, 224640, 13557351, 3864, 460196, -24, 10486, 0, 148, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 148*x^14 + 10486*x^12 - 24*x^11 + 460196*x^10 + 3864*x^9 + 13557351*x^8 + 224640*x^7 + 272216812*x^6 + 2718048*x^5 + 3610023514*x^4 - 36903144*x^3 + 28731561716*x^2 - 747884040*x + 104415192286)
 
gp: K = bnfinit(x^16 + 148*x^14 + 10486*x^12 - 24*x^11 + 460196*x^10 + 3864*x^9 + 13557351*x^8 + 224640*x^7 + 272216812*x^6 + 2718048*x^5 + 3610023514*x^4 - 36903144*x^3 + 28731561716*x^2 - 747884040*x + 104415192286, 1)
 

Normalized defining polynomial

\( x^{16} + 148 x^{14} + 10486 x^{12} - 24 x^{11} + 460196 x^{10} + 3864 x^{9} + 13557351 x^{8} + 224640 x^{7} + 272216812 x^{6} + 2718048 x^{5} + 3610023514 x^{4} - 36903144 x^{3} + 28731561716 x^{2} - 747884040 x + 104415192286 \)

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:  \(5032238723456334453905817600000000=2^{48}\cdot 3^{8}\cdot 5^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.75$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(3911,·)$, $\chi_{4080}(2891,·)$, $\chi_{4080}(1871,·)$, $\chi_{4080}(851,·)$, $\chi_{4080}(2041,·)$, $\chi_{4080}(3739,·)$, $\chi_{4080}(2719,·)$, $\chi_{4080}(1699,·)$, $\chi_{4080}(679,·)$, $\chi_{4080}(1021,·)$, $\chi_{4080}(3569,·)$, $\chi_{4080}(3061,·)$, $\chi_{4080}(1529,·)$, $\chi_{4080}(509,·)$, $\chi_{4080}(2549,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{20600506409} a^{14} - \frac{6746824366}{20600506409} a^{13} - \frac{4038700931}{20600506409} a^{12} + \frac{8014398951}{20600506409} a^{11} - \frac{6058036679}{20600506409} a^{10} + \frac{6375823592}{20600506409} a^{9} - \frac{300649151}{2942929487} a^{8} + \frac{2626363545}{20600506409} a^{7} - \frac{3792147596}{20600506409} a^{6} + \frac{930534973}{20600506409} a^{5} + \frac{8264567614}{20600506409} a^{4} + \frac{1185825177}{20600506409} a^{3} + \frac{9903741304}{20600506409} a^{2} + \frac{3765399070}{20600506409} a - \frac{3030760595}{20600506409}$, $\frac{1}{3026045400324734961865019277486009511598483609} a^{15} + \frac{323042268215821886433476260722431}{3026045400324734961865019277486009511598483609} a^{14} - \frac{736751227040728196297926907865592266191263617}{3026045400324734961865019277486009511598483609} a^{13} + \frac{38604755040470767261352639592047428658568932}{432292200046390708837859896783715644514069087} a^{12} - \frac{1166620651773410626057587229891723992787624071}{3026045400324734961865019277486009511598483609} a^{11} + \frac{627479886387654980946896732781644660177467232}{3026045400324734961865019277486009511598483609} a^{10} - \frac{510440313289961899596676221132714941413086129}{3026045400324734961865019277486009511598483609} a^{9} - \frac{157186841149099003437664464458755570247182716}{3026045400324734961865019277486009511598483609} a^{8} + \frac{664325078571085729069116679625327533141606197}{3026045400324734961865019277486009511598483609} a^{7} - \frac{1155609342163267672269514582111120896995324849}{3026045400324734961865019277486009511598483609} a^{6} - \frac{1164502078480570959179153468774046731409580888}{3026045400324734961865019277486009511598483609} a^{5} + \frac{19693972799911420941596630197983654771224734}{432292200046390708837859896783715644514069087} a^{4} + \frac{162058160653801980639834181839854815170073559}{432292200046390708837859896783715644514069087} a^{3} - \frac{18597009537992607777266177059515695925278673}{64383944687760318337553601648638500246776247} a^{2} + \frac{1330122179750534892025537667339373693235312873}{3026045400324734961865019277486009511598483609} a - \frac{1213212550720481542905210206433220692008069809}{3026045400324734961865019277486009511598483609}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{24}\times C_{6240}$, which has order $4792320$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-510}) \), \(\Q(\sqrt{-170}) \), \(\Q(\sqrt{3}, \sqrt{-85})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-170})\), \(\Q(\sqrt{2}, \sqrt{-255})\), \(\Q(\sqrt{6}, \sqrt{-170})\), \(\Q(\sqrt{2}, \sqrt{-85})\), \(\Q(\sqrt{6}, \sqrt{-85})\), 4.4.18432.1, \(\Q(\zeta_{16})^+\), 4.0.14796800.5, 4.0.133171200.5, 8.0.277102632960000.225, \(\Q(\zeta_{48})^+\), 8.0.70938274037760000.16, 8.0.17734568509440000.260, 8.0.17734568509440000.473, 8.0.70938274037760000.13, 8.0.875781160960000.56

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 R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$