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Properties

Label 16.8.90495174703...3409.4
Degree $16$
Signature $[8, 4]$
Discriminant $29^{10}\cdot 173^{14}$
Root discriminant $745.23$
Ramified primes $29, 173$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![284965015118076, -274349832137406, 89221420987155, -5114937956805, -3545496769300, 756003618333, 3690014449, -17659077219, 1582675217, 175767416, -29621345, -877763, 227127, 2345, -786, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 786*x^14 + 2345*x^13 + 227127*x^12 - 877763*x^11 - 29621345*x^10 + 175767416*x^9 + 1582675217*x^8 - 17659077219*x^7 + 3690014449*x^6 + 756003618333*x^5 - 3545496769300*x^4 - 5114937956805*x^3 + 89221420987155*x^2 - 274349832137406*x + 284965015118076)
 
gp: K = bnfinit(x^16 - 3*x^15 - 786*x^14 + 2345*x^13 + 227127*x^12 - 877763*x^11 - 29621345*x^10 + 175767416*x^9 + 1582675217*x^8 - 17659077219*x^7 + 3690014449*x^6 + 756003618333*x^5 - 3545496769300*x^4 - 5114937956805*x^3 + 89221420987155*x^2 - 274349832137406*x + 284965015118076, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 786 x^{14} + 2345 x^{13} + 227127 x^{12} - 877763 x^{11} - 29621345 x^{10} + 175767416 x^{9} + 1582675217 x^{8} - 17659077219 x^{7} + 3690014449 x^{6} + 756003618333 x^{5} - 3545496769300 x^{4} - 5114937956805 x^{3} + 89221420987155 x^{2} - 274349832137406 x + 284965015118076 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9049517470391542347782977198790354514633273409=29^{10}\cdot 173^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $745.23$
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:  $29, 173$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{11} + \frac{1}{18} a^{7} - \frac{1}{2} a^{6} - \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{18} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{11} + \frac{1}{18} a^{8} - \frac{4}{9} a^{7} + \frac{1}{18} a^{6} + \frac{4}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{2} a^{3} + \frac{4}{9} a^{2} - \frac{1}{2} a$, $\frac{1}{11934} a^{14} - \frac{149}{11934} a^{13} - \frac{59}{11934} a^{12} - \frac{209}{3978} a^{11} - \frac{19}{1989} a^{10} + \frac{955}{11934} a^{9} + \frac{965}{11934} a^{8} + \frac{151}{918} a^{7} + \frac{17}{78} a^{6} + \frac{430}{1989} a^{5} + \frac{3295}{11934} a^{4} + \frac{337}{918} a^{3} - \frac{1409}{3978} a^{2} - \frac{37}{1326} a + \frac{45}{221}$, $\frac{1}{44868189086151065854280286047968883327965023681104012172928868449888025876157094047131196} a^{15} - \frac{2019934210066060419399442173162203857714939005907320148388631058942289367279174245}{131193535339622999573918965052540594526213519535391848458856340496748613672973959202138} a^{14} - \frac{111723745636917789380941282806625442943903337475618623858514436271655903884652192403353}{7478031514358510975713381007994813887994170613517335362154811408314670979359515674521866} a^{13} + \frac{357326897791656294808448417443249250342696621029656555190626418206730102056640715670191}{44868189086151065854280286047968883327965023681104012172928868449888025876157094047131196} a^{12} + \frac{17534927905390502478865597720746146227670833316482876693491554814552613214809250069957}{131193535339622999573918965052540594526213519535391848458856340496748613672973959202138} a^{11} + \frac{3208505490084215288512332641303005358508340913882734166001145154985263922842616772555873}{44868189086151065854280286047968883327965023681104012172928868449888025876157094047131196} a^{10} - \frac{541158785706697398442336991088597550255802665489307838387738069767881924631067813349196}{11217047271537766463570071511992220831991255920276003043232217112472006469039273511782799} a^{9} - \frac{2574070113502009282469594653854704792503126770187425918264979205688689671529486248826429}{22434094543075532927140143023984441663982511840552006086464434224944012938078547023565598} a^{8} - \frac{9419288128743059883464561196060203375147127058073321326044156019954420301653510058219653}{44868189086151065854280286047968883327965023681104012172928868449888025876157094047131196} a^{7} - \frac{1445636552201578051012005487778611660445401498090514005206125032276199785188656531553253}{3739015757179255487856690503997406943997085306758667681077405704157335489679757837260933} a^{6} + \frac{6923306475113318419188841829892479028085188659332705419543100707804501390656589162407971}{44868189086151065854280286047968883327965023681104012172928868449888025876157094047131196} a^{5} - \frac{1064789319055311592255672511408891398934749768632251522961487130953376931498678323732546}{3739015757179255487856690503997406943997085306758667681077405704157335489679757837260933} a^{4} - \frac{1486089862033403610060254617174607106566197013619454955438914102282004594478237954363893}{22434094543075532927140143023984441663982511840552006086464434224944012938078547023565598} a^{3} + \frac{38994718985773953417985825678961259208308622329897938391522663200800426435760684908889}{107597575746165625549832820258918185438765044798810580750428941126829798264165693158588} a^{2} + \frac{65854646835800342287184797399589594163719240250648204361220963587916107601624754139194}{138482065080713166216914463111015071999892048398469173373237248302123536654805845824479} a + \frac{40488912218846472704918488201960569922110535547924744957342092840142063751931259517395}{415446195242139498650743389333045215999676145195407520119711744906370609964417537473437}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 214265958934000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5017}) \), \(\Q(\sqrt{173}) \), \(\Q(\sqrt{29}) \), 4.4.4354459997.1 x2, 4.4.150153793.1 x2, \(\Q(\sqrt{29}, \sqrt{173})\), 8.8.18961321865473240009.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$173$173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$