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Properties

Label 16.0.67876862126...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 101^{10}\cdot 673^{8}\cdot 1229^{4}$
Root discriminant $15{,}030.82$
Ramified primes $2, 5, 101, 673, 1229$
Class number Not computed
Class group Not computed
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45933415411470942405369606400, 0, 591998115627081544732705600, 0, 2340628252707564433358240, 0, 2989272107913514337760, 0, 788070267257863489, 0, 83745030765827, 0, 4286485093, 0, 105511, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 105511*x^14 + 4286485093*x^12 + 83745030765827*x^10 + 788070267257863489*x^8 + 2989272107913514337760*x^6 + 2340628252707564433358240*x^4 + 591998115627081544732705600*x^2 + 45933415411470942405369606400)
 
gp: K = bnfinit(x^16 + 105511*x^14 + 4286485093*x^12 + 83745030765827*x^10 + 788070267257863489*x^8 + 2989272107913514337760*x^6 + 2340628252707564433358240*x^4 + 591998115627081544732705600*x^2 + 45933415411470942405369606400, 1)
 

Normalized defining polynomial

\( x^{16} + 105511 x^{14} + 4286485093 x^{12} + 83745030765827 x^{10} + 788070267257863489 x^{8} + 2989272107913514337760 x^{6} + 2340628252707564433358240 x^{4} + 591998115627081544732705600 x^{2} + 45933415411470942405369606400 \)

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:  \(6787686212645735395617770359527756187429905390010541799040000000000=2^{16}\cdot 5^{10}\cdot 101^{10}\cdot 673^{8}\cdot 1229^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15{,}030.82$
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:  $2, 5, 101, 673, 1229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{172060} a^{10} + \frac{10287}{34412} a^{8} - \frac{21107}{172060} a^{6} - \frac{60601}{172060} a^{4} + \frac{5057}{34412} a^{2} + \frac{2}{7}$, $\frac{1}{344120} a^{11} + \frac{10287}{68824} a^{9} + \frac{150953}{344120} a^{7} - \frac{60601}{344120} a^{5} + \frac{5057}{68824} a^{3} - \frac{5}{14} a$, $\frac{1}{598013800720} a^{12} + \frac{1595059}{598013800720} a^{10} - \frac{42490553601}{85430542960} a^{8} - \frac{289041580049}{598013800720} a^{6} - \frac{134961007099}{598013800720} a^{4} - \frac{12997}{240884} a^{2} - \frac{3}{49}$, $\frac{1}{1196027601440} a^{13} + \frac{1595059}{1196027601440} a^{11} - \frac{42490553601}{170861085920} a^{9} + \frac{308972220671}{1196027601440} a^{7} - \frac{134961007099}{1196027601440} a^{5} + \frac{227887}{481768} a^{3} + \frac{23}{49} a$, $\frac{1}{11310633498098506966763623547343279304161244444172572332852636872842370872342444480} a^{14} - \frac{1217115399995404629509320715830883489331776216292087388320555380531877}{11310633498098506966763623547343279304161244444172572332852636872842370872342444480} a^{12} + \frac{8706764449309806278962850568558191590860155078852003217321060540447856167937}{11310633498098506966763623547343279304161244444172572332852636872842370872342444480} a^{10} - \frac{1686640388248289844440887448526432704285469629483157762787995286738502227092701737}{11310633498098506966763623547343279304161244444172572332852636872842370872342444480} a^{8} + \frac{3975970937336823417434388016895538248663209191352848898266083609658546160579473549}{11310633498098506966763623547343279304161244444172572332852636872842370872342444480} a^{6} + \frac{14202266961987940063504365471246405476612381712446960901967486637960315852861}{2300779800264138927331900640224426221350944760816227081540406198706747533023280} a^{4} + \frac{287341445501740808764179650821840016819839468555530767143921752182048341}{926769651034060907335071031034015508604332895945438649123253308536581916} a^{2} - \frac{7824478130739969206264281057899403591233820236082383629823785429106}{188521084425154781801275636906837979781190580949031458324502300353251}$, $\frac{1}{397252069720215761686671986229790655720751227368229085474450312247969749778411335026560} a^{15} - \frac{19394316895943796109117367889087645820022008222640254148314071102370210389}{79450413944043152337334397245958131144150245473645817094890062449593949955682267005312} a^{13} + \frac{11122868317147701535787373809044772421645553477266206068767933797433744538015701}{397252069720215761686671986229790655720751227368229085474450312247969749778411335026560} a^{11} + \frac{43837554036369892111937648253955188886449175423610221436611477333268247485791948693459}{397252069720215761686671986229790655720751227368229085474450312247969749778411335026560} a^{9} + \frac{163292016657558568958365503624192994721196434671259652199877585388835323793579587861489}{397252069720215761686671986229790655720751227368229085474450312247969749778411335026560} a^{7} - \frac{6397428468477621175047137443712569213297374543703266806874523007487335796740059757}{20201997036219271851437753571490574436571970472346881889465536627744596713710910040} a^{5} + \frac{333303186390213206919833798289622443664043771643898125454121542756966491725}{8137500920904571796855591187994173173300344992848924058626725675605457513438} a^{3} + \frac{2062501328621670298650655924697073217229877954075674759921965414587465911}{6621237527180286246424402919441963525874975584091882879273169793006881622} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{339865}) \), \(\Q(\sqrt{3365}) \), 4.4.51005.1, 4.4.23101643645.1, \(\Q(\sqrt{101}, \sqrt{3365})\), 8.8.13342148477514222150625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
673Data not computed
1229Data not computed