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Properties

Label 16.0.67876862126...0000.1
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, 675683468660538284973587200, 0, 1723001890065898798219360, 0, 1226168211290116622980, 0, 358242690060063549, 0, 49805057274891, 0, 3348619917, 0, 99927, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 99927*x^14 + 3348619917*x^12 + 49805057274891*x^10 + 358242690060063549*x^8 + 1226168211290116622980*x^6 + 1723001890065898798219360*x^4 + 675683468660538284973587200*x^2 + 45933415411470942405369606400)
 
gp: K = bnfinit(x^16 + 99927*x^14 + 3348619917*x^12 + 49805057274891*x^10 + 358242690060063549*x^8 + 1226168211290116622980*x^6 + 1723001890065898798219360*x^4 + 675683468660538284973587200*x^2 + 45933415411470942405369606400, 1)
 

Normalized defining polynomial

\( x^{16} + 99927 x^{14} + 3348619917 x^{12} + 49805057274891 x^{10} + 358242690060063549 x^{8} + 1226168211290116622980 x^{6} + 1723001890065898798219360 x^{4} + 675683468660538284973587200 x^{2} + 45933415411470942405369606400 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{24580} a^{10} + \frac{6523}{24580} a^{8} + \frac{589}{4916} a^{6} + \frac{8811}{24580} a^{4} - \frac{475}{4916} a^{2}$, $\frac{1}{49160} a^{11} + \frac{6523}{49160} a^{9} + \frac{589}{9832} a^{7} - \frac{15769}{49160} a^{5} - \frac{475}{9832} a^{3}$, $\frac{1}{12204363280} a^{12} + \frac{99927}{12204363280} a^{10} + \frac{3348619917}{12204363280} a^{8} - \frac{949270789}{12204363280} a^{6} - \frac{3564451411}{12204363280} a^{4} - \frac{1267}{4916} a^{2}$, $\frac{1}{24408726560} a^{13} + \frac{99927}{24408726560} a^{11} + \frac{3348619917}{24408726560} a^{9} + \frac{11255092491}{24408726560} a^{7} + \frac{8639911869}{24408726560} a^{5} - \frac{1267}{9832} a^{3}$, $\frac{1}{208599805219007339812338739322474402301203928623868731160927589531426484121737374984640} a^{14} - \frac{2862446247601774516479296008316782191862716491095002958940031997870376621669}{208599805219007339812338739322474402301203928623868731160927589531426484121737374984640} a^{12} + \frac{442151078053620707163828191715051032466195952057264244919027478351791084094101689}{208599805219007339812338739322474402301203928623868731160927589531426484121737374984640} a^{10} - \frac{85618141569461722580801090189306059379155235393213563540165361239956983741612976395409}{208599805219007339812338739322474402301203928623868731160927589531426484121737374984640} a^{8} - \frac{86848741589836342407602377860170243197532591431705004971478727416975770651746201268487}{208599805219007339812338739322474402301203928623868731160927589531426484121737374984640} a^{6} + \frac{1425787544521407667889742573811200493923948668077551144110415529115304660334805861}{4243283263201939377793709099318030966257199524488786231914719071021694144054869304} a^{4} - \frac{1085663950929338622738002641044822903426518385584091436451135935764536749205}{4273057930864201131276443356627007957706498405377456347745811888259083437447} a^{2} - \frac{603954524043559437346184748877881637469480808180160698487538515465152662}{3476857551557527364748936823943863269085840850591909151949399420878017443}$, $\frac{1}{7326442358901975788888961202483945957622884381127517575834098799522760975323660084210526080} a^{15} + \frac{29720528520843781294642460692689150571614540191314491351960927462732314014259663}{7326442358901975788888961202483945957622884381127517575834098799522760975323660084210526080} a^{13} + \frac{2317145817713495592478778023147310102492669892549073012218129889686914028527243091637}{7326442358901975788888961202483945957622884381127517575834098799522760975323660084210526080} a^{11} + \frac{718063182979783149275077241285432219817523242449954304615957640486411637917296763726050307}{7326442358901975788888961202483945957622884381127517575834098799522760975323660084210526080} a^{9} - \frac{96285646096749799632733764989932100984415560829349134588712110814534181069994059920299775}{1465288471780395157777792240496789191524576876225503515166819759904552195064732016842105216} a^{7} + \frac{662030081379641466166754797569387904125199111465164482273451753660163045398557313257307}{1490325947701785148268706509862478835968853616990951500373087632124239417274951196950880} a^{5} - \frac{51729693232694979038312967178082866169070849455192644256010163806888678454609005}{300156681295624944265382487142907546981135273987334043691056810278871056980027068} a^{3} + \frac{1365801063238064694908985987061060383113265973474440136017626433889595702437}{122114190925803476104712159130556365736832902354489033234766806460077728633046} a$

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

\(\Q(\sqrt{339865}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{3365}) \), 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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