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Properties

Label 16.8.10998771473...7801.3
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 73^{14}$
Root discriminant $1005.97$
Ramified primes $37, 73$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9516392527011651, -13129043796800163, 7438959624205646, -2174612665157564, 320379262260514, -12394664061501, -2707188392044, 307874361192, 4616835965, -1683482107, -6123213, 6385537, -138630, 6072, -663, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 663*x^14 + 6072*x^13 - 138630*x^12 + 6385537*x^11 - 6123213*x^10 - 1683482107*x^9 + 4616835965*x^8 + 307874361192*x^7 - 2707188392044*x^6 - 12394664061501*x^5 + 320379262260514*x^4 - 2174612665157564*x^3 + 7438959624205646*x^2 - 13129043796800163*x + 9516392527011651)
 
gp: K = bnfinit(x^16 - 6*x^15 - 663*x^14 + 6072*x^13 - 138630*x^12 + 6385537*x^11 - 6123213*x^10 - 1683482107*x^9 + 4616835965*x^8 + 307874361192*x^7 - 2707188392044*x^6 - 12394664061501*x^5 + 320379262260514*x^4 - 2174612665157564*x^3 + 7438959624205646*x^2 - 13129043796800163*x + 9516392527011651, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 663 x^{14} + 6072 x^{13} - 138630 x^{12} + 6385537 x^{11} - 6123213 x^{10} - 1683482107 x^{9} + 4616835965 x^{8} + 307874361192 x^{7} - 2707188392044 x^{6} - 12394664061501 x^{5} + 320379262260514 x^{4} - 2174612665157564 x^{3} + 7438959624205646 x^{2} - 13129043796800163 x + 9516392527011651 \)

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:  \(1099877147396767625299704166328746389549375427801=37^{14}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1005.97$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{111} a^{8} - \frac{1}{37} a^{7} - \frac{1}{37} a^{6} - \frac{44}{111} a^{5} - \frac{7}{37} a^{4} + \frac{47}{111} a^{3} + \frac{49}{111} a^{2} + \frac{4}{111} a - \frac{7}{37}$, $\frac{1}{333} a^{9} - \frac{4}{111} a^{7} - \frac{16}{333} a^{6} + \frac{23}{111} a^{5} - \frac{164}{333} a^{4} - \frac{106}{333} a^{3} - \frac{34}{333} a^{2} + \frac{28}{333} a - \frac{7}{37}$, $\frac{1}{333} a^{10} - \frac{52}{333} a^{7} + \frac{11}{111} a^{6} - \frac{26}{333} a^{5} - \frac{25}{333} a^{4} - \frac{136}{333} a^{3} - \frac{50}{333} a^{2} - \frac{5}{111} a + \frac{9}{37}$, $\frac{1}{999} a^{11} + \frac{1}{999} a^{10} - \frac{1}{999} a^{9} - \frac{4}{999} a^{8} + \frac{71}{999} a^{7} - \frac{121}{999} a^{6} + \frac{70}{333} a^{5} + \frac{8}{37} a^{4} + \frac{67}{999} a^{3} - \frac{454}{999} a^{2} - \frac{103}{999} a + \frac{5}{37}$, $\frac{1}{999} a^{12} + \frac{1}{999} a^{10} + \frac{1}{333} a^{8} + \frac{55}{333} a^{7} - \frac{68}{999} a^{6} - \frac{1}{37} a^{5} - \frac{203}{999} a^{4} + \frac{31}{999} a^{3} + \frac{26}{111} a^{2} + \frac{322}{999} a + \frac{16}{37}$, $\frac{1}{999} a^{13} - \frac{1}{999} a^{10} + \frac{1}{999} a^{9} - \frac{2}{999} a^{8} + \frac{77}{999} a^{7} - \frac{11}{999} a^{6} + \frac{244}{999} a^{5} + \frac{235}{999} a^{4} + \frac{440}{999} a^{3} + \frac{491}{999} a^{2} + \frac{100}{999} a - \frac{13}{37}$, $\frac{1}{8991} a^{14} + \frac{2}{8991} a^{13} + \frac{2}{8991} a^{12} - \frac{1}{2997} a^{11} + \frac{11}{8991} a^{10} + \frac{5}{8991} a^{9} + \frac{2}{2997} a^{8} - \frac{419}{8991} a^{7} - \frac{746}{8991} a^{6} - \frac{328}{999} a^{5} + \frac{368}{999} a^{4} + \frac{845}{2997} a^{3} - \frac{4211}{8991} a^{2} + \frac{68}{999} a + \frac{18}{37}$, $\frac{1}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{15} - \frac{3171452363620800748771992258698218884086154497394189208862019563492857930834092581}{286388285885821181434650121264844199851927672704329270809362104688476571792908441662061} a^{14} - \frac{16133298153416105613052260781558177547760295699491811846480013970523537668372305549495}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{13} - \frac{6469444334541566889820237387097834744941940990268960447532595268104152620812525875492}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{12} - \frac{2687773799540789722516348178077021770430457097794449628114670728430783397573847669284}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{11} + \frac{1055273060264439926423361760246803716688690437246709520146215068293662538142211398468}{952047544971783927471944997718265853561813614665743251609501050721151846771019954714419} a^{10} - \frac{50082175621087568907240758996071466963762260772620313803784645246025270469206205992964}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{9} + \frac{33617735678434697731068718051293975049750052925135114555898922215387133667502471124617}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{8} + \frac{3877411834425897848757530766980064878288419334994965090956081844312393403743337505227281}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{7} - \frac{5092560942187952936094965027250883442898732170189812083905926636220026136983862588347241}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{6} + \frac{16091630932236316156391403760076802121706182767914404133251770943431691003348662924470}{434885915604395127363727961920689340515890169662129633451253566378797757167009115116463} a^{5} + \frac{2564895566855413689675267994538681244282175107469271501917843364519081881302849446561117}{11741919721318668438820654971858612193929034580877500103183846292227539443509246108144501} a^{4} + \frac{15772133654297817936341453210852468633431081604228900142105385053724049382436628347225018}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{3} + \frac{14441118860024124860425698195729835498584847112789563464067850656882395534072291074112886}{35225759163956005316461964915575836581787103742632500309551538876682618330527738324433503} a^{2} - \frac{81117456597562287181094631294260300085287237595151949710656921702189955319797391930846}{1304657746813185382091183885762068021547670508986388900353760699136393271501027345349389} a + \frac{72107275135054728733844857699420240133317535805613386117249313903047750091647748270549}{144961971868131709121242653973563113505296723220709877817084522126265919055669705038821}$

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

Class group and class number

$C_{4}$, 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:  $11$
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:  \( 496887369393000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{2701}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{37}, \sqrt{73})\), 8.8.388282220975269366201.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$73$73.8.7.4$x^{8} - 1140625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.4$x^{8} - 1140625$$8$$1$$7$$C_8$$[\ ]_{8}$