↑

Properties

Label 16.0.22138542742...0625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{6}\cdot 661^{4}$
Root discriminant $68.25$
Ramified primes $5, 41, 661$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T864

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6388889431, 2395549837, 3374280365, 764606308, 681059894, 90460561, 72089534, 4604426, 4848771, 57448, 224236, -3233, 6904, -116, 125, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 125*x^14 - 116*x^13 + 6904*x^12 - 3233*x^11 + 224236*x^10 + 57448*x^9 + 4848771*x^8 + 4604426*x^7 + 72089534*x^6 + 90460561*x^5 + 681059894*x^4 + 764606308*x^3 + 3374280365*x^2 + 2395549837*x + 6388889431)
 
gp: K = bnfinit(x^16 - x^15 + 125*x^14 - 116*x^13 + 6904*x^12 - 3233*x^11 + 224236*x^10 + 57448*x^9 + 4848771*x^8 + 4604426*x^7 + 72089534*x^6 + 90460561*x^5 + 681059894*x^4 + 764606308*x^3 + 3374280365*x^2 + 2395549837*x + 6388889431, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 125 x^{14} - 116 x^{13} + 6904 x^{12} - 3233 x^{11} + 224236 x^{10} + 57448 x^{9} + 4848771 x^{8} + 4604426 x^{7} + 72089534 x^{6} + 90460561 x^{5} + 681059894 x^{4} + 764606308 x^{3} + 3374280365 x^{2} + 2395549837 x + 6388889431 \)

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:  \(221385427428590205586181640625=5^{12}\cdot 41^{6}\cdot 661^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.25$
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:  $5, 41, 661$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{49204479599952480705787008783896818474371001207993963252042788834} a^{15} + \frac{3222273743917143303407385249888604382866072913757735051674980757}{49204479599952480705787008783896818474371001207993963252042788834} a^{14} + \frac{117927460201067712597094641835663000547033050378507666684260962}{8200746599992080117631168130649469745728500201332327208673798139} a^{13} - \frac{2754527509216782767663355349699868832777075191306135524802785952}{24602239799976240352893504391948409237185500603996981626021394417} a^{12} - \frac{4858608662833050206923364543558867851848563253058433566053032287}{16401493199984160235262336261298939491457000402664654417347596278} a^{11} + \frac{1632461494956334358883472493751303707613482608395569835895561826}{24602239799976240352893504391948409237185500603996981626021394417} a^{10} - \frac{10679227313140272322585597362950886533409745134016818669889481179}{24602239799976240352893504391948409237185500603996981626021394417} a^{9} + \frac{19437049026865548619052894154660054565954708262704173068223456929}{49204479599952480705787008783896818474371001207993963252042788834} a^{8} + \frac{1738523413795842937059562902619118098680435568749577121053765377}{8200746599992080117631168130649469745728500201332327208673798139} a^{7} - \frac{32454971000913367524449507505410978735917883031885293665385599}{102083982572515520136487570091072237498695023253099508821665537} a^{6} - \frac{3137110251395485846999616333973624744910679439717785593029085547}{16401493199984160235262336261298939491457000402664654417347596278} a^{5} + \frac{10421984354387659456213910203394482334454063830494203675798577202}{24602239799976240352893504391948409237185500603996981626021394417} a^{4} - \frac{2205365265079559886255997152797486500753359713634489018477264899}{24602239799976240352893504391948409237185500603996981626021394417} a^{3} + \frac{13277790580159612726200347701110130141435376145065487401869075619}{49204479599952480705787008783896818474371001207993963252042788834} a^{2} - \frac{1844960163209007185768816744700467547356953551972420131312518004}{8200746599992080117631168130649469745728500201332327208673798139} a + \frac{6224264844881716502063247127145083796412669933562774431804804935}{49204479599952480705787008783896818474371001207993963252042788834}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -\frac{1180191652548609825051522286815776602405215771745}{479120188733115028648408916934333514805048322009426720054} a^{15} + \frac{5385369487955005936628700346083087189766680488207}{479120188733115028648408916934333514805048322009426720054} a^{14} - \frac{141924220762071788757524338502537603574544303309835}{479120188733115028648408916934333514805048322009426720054} a^{13} + \frac{103259195851869148583239280993476483652744020991266}{79853364788852504774734819489055585800841387001571120009} a^{12} - \frac{2502720221237101986523571207428280865599593077447377}{159706729577705009549469638978111171601682774003142240018} a^{11} + \frac{28005948949769340822748506638095002774309195176299037}{479120188733115028648408916934333514805048322009426720054} a^{10} - \frac{36973699520395956844401713906509370510514834404491678}{79853364788852504774734819489055585800841387001571120009} a^{9} + \frac{662333597195656369897054338804099490535099588932619161}{479120188733115028648408916934333514805048322009426720054} a^{8} - \frac{3910732149546256714525863433613277189039932678925577291}{479120188733115028648408916934333514805048322009426720054} a^{7} + \frac{1661521103071068412626218656196983524711891440606514039}{79853364788852504774734819489055585800841387001571120009} a^{6} - \frac{13261122103929107531349089659677635374568017082978269343}{159706729577705009549469638978111171601682774003142240018} a^{5} + \frac{115255690917279643397511867597055241852532146561984068175}{479120188733115028648408916934333514805048322009426720054} a^{4} - \frac{95388400629484284696165200872683841207703643832065688814}{239560094366557514324204458467166757402524161004713360027} a^{3} + \frac{891698191435822268249390798957253408478442195207521778819}{479120188733115028648408916934333514805048322009426720054} a^{2} - \frac{148434372331418421321732916570067308417463615885498231465}{479120188733115028648408916934333514805048322009426720054} a + \frac{922183963740866477423983282279876938560797568048455189241}{159706729577705009549469638978111171601682774003142240018} \) (order $10$)
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:  \( 156301284.289 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T864:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n864
Character table for t16n864 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.26265625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$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}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
661Data not computed