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Properties

Label 16.0.76573533698...000.81
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $201.96$
Ramified primes $2, 3, 5, 19$
Class number $11141120$ (GRH)
Class group $[2, 4, 4, 16, 16, 1360]$ (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![1625086216321, 129839059720, 297775783704, 15549031600, 24866069920, 761060200, 1261661416, 18874400, 43290174, 203000, 1038536, -1040, 17200, -40, 184, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 184*x^14 - 40*x^13 + 17200*x^12 - 1040*x^11 + 1038536*x^10 + 203000*x^9 + 43290174*x^8 + 18874400*x^7 + 1261661416*x^6 + 761060200*x^5 + 24866069920*x^4 + 15549031600*x^3 + 297775783704*x^2 + 129839059720*x + 1625086216321)
 
gp: K = bnfinit(x^16 + 184*x^14 - 40*x^13 + 17200*x^12 - 1040*x^11 + 1038536*x^10 + 203000*x^9 + 43290174*x^8 + 18874400*x^7 + 1261661416*x^6 + 761060200*x^5 + 24866069920*x^4 + 15549031600*x^3 + 297775783704*x^2 + 129839059720*x + 1625086216321, 1)
 

Normalized defining polynomial

\( x^{16} + 184 x^{14} - 40 x^{13} + 17200 x^{12} - 1040 x^{11} + 1038536 x^{10} + 203000 x^{9} + 43290174 x^{8} + 18874400 x^{7} + 1261661416 x^{6} + 761060200 x^{5} + 24866069920 x^{4} + 15549031600 x^{3} + 297775783704 x^{2} + 129839059720 x + 1625086216321 \)

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:  \(7657353369870241665908736000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $201.96$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4560=2^{4}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4560}(1,·)$, $\chi_{4560}(1027,·)$, $\chi_{4560}(3077,·)$, $\chi_{4560}(3649,·)$, $\chi_{4560}(1483,·)$, $\chi_{4560}(3533,·)$, $\chi_{4560}(911,·)$, $\chi_{4560}(1369,·)$, $\chi_{4560}(4559,·)$, $\chi_{4560}(797,·)$, $\chi_{4560}(1253,·)$, $\chi_{4560}(2279,·)$, $\chi_{4560}(2281,·)$, $\chi_{4560}(3307,·)$, $\chi_{4560}(3763,·)$, $\chi_{4560}(3191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{24} a^{10} + \frac{1}{24} a^{9} - \frac{1}{12} a^{8} - \frac{1}{4} a^{7} + \frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} - \frac{1}{4} a^{3} - \frac{11}{24} a^{2} - \frac{11}{24} a + \frac{1}{6}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{8} + \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{12} a^{4} - \frac{11}{24} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{11}{24}$, $\frac{1}{72} a^{12} - \frac{1}{72} a^{10} + \frac{1}{72} a^{9} + \frac{1}{72} a^{8} + \frac{1}{12} a^{7} + \frac{5}{36} a^{6} - \frac{5}{36} a^{5} + \frac{17}{72} a^{4} - \frac{1}{4} a^{3} + \frac{23}{72} a^{2} + \frac{25}{72} a - \frac{7}{72}$, $\frac{1}{144} a^{13} - \frac{1}{144} a^{12} + \frac{1}{72} a^{11} + \frac{1}{72} a^{10} - \frac{1}{16} a^{9} - \frac{7}{144} a^{8} + \frac{7}{36} a^{7} - \frac{5}{36} a^{6} - \frac{1}{16} a^{5} - \frac{23}{144} a^{4} + \frac{13}{72} a^{3} - \frac{35}{72} a^{2} + \frac{49}{144} a + \frac{31}{144}$, $\frac{1}{325141740144} a^{14} - \frac{513374353}{325141740144} a^{13} - \frac{96513431}{81285435036} a^{12} - \frac{349345309}{20321358759} a^{11} - \frac{3406840103}{325141740144} a^{10} - \frac{14327953631}{325141740144} a^{9} + \frac{6309493051}{54190290024} a^{8} - \frac{3555172975}{40642717518} a^{7} + \frac{169792067}{325141740144} a^{6} - \frac{50055792763}{325141740144} a^{5} + \frac{2372325949}{27095145012} a^{4} + \frac{9852570329}{20321358759} a^{3} - \frac{23749669597}{325141740144} a^{2} - \frac{1142571113}{2778989232} a + \frac{14207844695}{162570870072}$, $\frac{1}{40460783776874861973060312688665291624780278256} a^{15} - \frac{58543515466458308233874265302547137}{40460783776874861973060312688665291624780278256} a^{14} - \frac{305412849174441458330546259448311079334965}{10115195944218715493265078172166322906195069564} a^{13} - \frac{124297109593130221205730459286632794962219273}{20230391888437430986530156344332645812390139128} a^{12} - \frac{63626857791190059957081328970882254586602207}{3112367982836527844081562514512714740367713712} a^{11} + \frac{82130928404339011905422674681187849818201727}{13486927925624953991020104229555097208260092752} a^{10} + \frac{501575790511106006332266359581319689916868907}{20230391888437430986530156344332645812390139128} a^{9} + \frac{2140711554357714544488374343562377572151826763}{20230391888437430986530156344332645812390139128} a^{8} + \frac{701780782873612554910632025306188404189825131}{40460783776874861973060312688665291624780278256} a^{7} - \frac{321826056919025935552418981473483824731815675}{4495642641874984663673368076518365736086697584} a^{6} - \frac{15691411125665019655108063689958231135592375}{10115195944218715493265078172166322906195069564} a^{5} - \frac{3500810634427632082323854761896890644852041923}{20230391888437430986530156344332645812390139128} a^{4} + \frac{745616063788744855729407923096693675590449181}{13486927925624953991020104229555097208260092752} a^{3} - \frac{8217268214779210715802814397204630111492245025}{40460783776874861973060312688665291624780278256} a^{2} + \frac{1501100394546444472565842808501422677309695669}{20230391888437430986530156344332645812390139128} a - \frac{6636509909772154303632328885175387459443263587}{20230391888437430986530156344332645812390139128}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{16}\times C_{16}\times C_{1360}$, which has order $11141120$ (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:  \( 85299.42553126559 \) (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

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-570}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-114}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{2}, \sqrt{-285})\), \(\Q(\sqrt{10}, \sqrt{-114})\), \(\Q(\sqrt{5}, \sqrt{-57})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-57})\), \(\Q(\sqrt{10}, \sqrt{-57})\), \(\Q(\sqrt{5}, \sqrt{-114})\), 4.0.831744000.2, 4.4.256000.1, 4.0.831744000.1, 4.4.256000.2, 8.0.432373800960000.208, 8.0.2767192326144000000.6, 8.0.2767192326144000000.8, 8.0.691798081536000000.23, 8.8.65536000000.1, 8.0.2767192326144000000.20, 8.0.2767192326144000000.19

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$