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Properties

Label 16.8.10998771473...7801.4
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![221064824525695641, 21965648393476953, -6759690646580883, -409151283885081, 143362781560899, 6006374183562, -2677277154937, -154648818706, 21309733002, 1268065652, -62583259, -5072721, 176560, 8110, -678, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 678*x^14 + 8110*x^13 + 176560*x^12 - 5072721*x^11 - 62583259*x^10 + 1268065652*x^9 + 21309733002*x^8 - 154648818706*x^7 - 2677277154937*x^6 + 6006374183562*x^5 + 143362781560899*x^4 - 409151283885081*x^3 - 6759690646580883*x^2 + 21965648393476953*x + 221064824525695641)
 
gp: K = bnfinit(x^16 - 2*x^15 - 678*x^14 + 8110*x^13 + 176560*x^12 - 5072721*x^11 - 62583259*x^10 + 1268065652*x^9 + 21309733002*x^8 - 154648818706*x^7 - 2677277154937*x^6 + 6006374183562*x^5 + 143362781560899*x^4 - 409151283885081*x^3 - 6759690646580883*x^2 + 21965648393476953*x + 221064824525695641, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 678 x^{14} + 8110 x^{13} + 176560 x^{12} - 5072721 x^{11} - 62583259 x^{10} + 1268065652 x^{9} + 21309733002 x^{8} - 154648818706 x^{7} - 2677277154937 x^{6} + 6006374183562 x^{5} + 143362781560899 x^{4} - 409151283885081 x^{3} - 6759690646580883 x^{2} + 21965648393476953 x + 221064824525695641 \)

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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{333} a^{8} - \frac{1}{333} a^{7} + \frac{4}{111} a^{6} - \frac{40}{333} a^{5} - \frac{1}{111} a^{4} + \frac{41}{333} a^{3} - \frac{28}{333} a^{2} - \frac{4}{37} a + \frac{7}{37}$, $\frac{1}{2997} a^{9} + \frac{53}{999} a^{7} + \frac{1}{333} a^{6} + \frac{109}{999} a^{5} - \frac{41}{333} a^{4} - \frac{172}{2997} a^{3} - \frac{77}{333} a^{2} - \frac{71}{333} a + \frac{9}{37}$, $\frac{1}{2997} a^{10} - \frac{1}{999} a^{8} - \frac{2}{37} a^{7} + \frac{16}{999} a^{6} - \frac{8}{111} a^{5} - \frac{19}{2997} a^{4} + \frac{4}{37} a^{3} - \frac{16}{111} a^{2} - \frac{16}{111} a - \frac{15}{37}$, $\frac{1}{8991} a^{11} + \frac{1}{8991} a^{10} + \frac{2}{2997} a^{8} + \frac{64}{2997} a^{7} - \frac{140}{2997} a^{6} - \frac{100}{8991} a^{5} + \frac{1349}{8991} a^{4} - \frac{202}{2997} a^{3} + \frac{103}{999} a^{2} - \frac{101}{333} a - \frac{1}{37}$, $\frac{1}{26973} a^{12} - \frac{1}{26973} a^{11} - \frac{2}{26973} a^{10} - \frac{1}{8991} a^{9} - \frac{1}{2997} a^{8} + \frac{317}{8991} a^{7} - \frac{610}{26973} a^{6} + \frac{172}{26973} a^{5} + \frac{1583}{26973} a^{4} - \frac{355}{8991} a^{3} - \frac{449}{2997} a^{2} - \frac{8}{999} a$, $\frac{1}{26973} a^{13} - \frac{2}{26973} a^{10} - \frac{1}{8991} a^{9} - \frac{4}{8991} a^{8} + \frac{323}{26973} a^{7} - \frac{431}{8991} a^{6} + \frac{1439}{8991} a^{5} - \frac{1834}{26973} a^{4} + \frac{875}{8991} a^{3} - \frac{1496}{2997} a^{2} + \frac{439}{999} a - \frac{2}{37}$, $\frac{1}{242757} a^{14} + \frac{1}{80919} a^{13} + \frac{1}{242757} a^{11} + \frac{13}{80919} a^{10} - \frac{7}{80919} a^{9} - \frac{235}{242757} a^{8} - \frac{404}{26973} a^{7} + \frac{3821}{80919} a^{6} - \frac{6679}{242757} a^{5} - \frac{2161}{26973} a^{4} - \frac{3712}{26973} a^{3} - \frac{11}{2997} a^{2} + \frac{482}{2997} a + \frac{35}{111}$, $\frac{1}{901578031212595274604037462800593152257195911748334947691775530027601476849575392789140509} a^{15} + \frac{400058342071692270720962076196664111153488828360388224014976457659759633589474032790}{901578031212595274604037462800593152257195911748334947691775530027601476849575392789140509} a^{14} + \frac{1879684935183099623848063693072941142312501249964862438161146953879819549141705523415}{300526010404198424868012487600197717419065303916111649230591843342533825616525130929713503} a^{13} + \frac{7754925080222184526283776751342538923621766681457241035560911984446322769735854439403}{901578031212595274604037462800593152257195911748334947691775530027601476849575392789140509} a^{12} + \frac{41344506753724902539717125936152486483657775312963813639697353438654585574747148153788}{901578031212595274604037462800593152257195911748334947691775530027601476849575392789140509} a^{11} + \frac{29633205199603589441330315361190824636365814895057959829184037741027211141612097688133}{300526010404198424868012487600197717419065303916111649230591843342533825616525130929713503} a^{10} - \frac{146243027318070848448886249545009746780058078837154215968817558567209005220666539859110}{901578031212595274604037462800593152257195911748334947691775530027601476849575392789140509} a^{9} + \frac{1308200903729755301287442378896600308552191090908032948777244547731125807619883069710276}{901578031212595274604037462800593152257195911748334947691775530027601476849575392789140509} a^{8} + \frac{851719506535814446923673499860151877949544234683358156693364335325094369917501532423}{59287040916196177721051980193371023361425390395760830386780793715236501403930781402587} a^{7} - \frac{15874381282774716833620283908075810302929118650196749925231561434002950852424820791406485}{901578031212595274604037462800593152257195911748334947691775530027601476849575392789140509} a^{6} + \frac{69792338164329220032568835500499745764938366195365372107548800842144117935224420475108519}{901578031212595274604037462800593152257195911748334947691775530027601476849575392789140509} a^{5} - \frac{19457856187538400418144031064981337164577914811569559470872195771381966704246539827473}{731206837966419525226307755718242621457579814881050241436963122487916850648479637298573} a^{4} - \frac{13456991882353262537794493392861616974592651923752437253594387939787861931559559040620286}{100175336801399474956004162533399239139688434638703883076863947780844608538841710309904501} a^{3} + \frac{22150277438470048053064172003841695174623207633515089493313601614592441425746247555694}{137414728122633024631007081664470835582563010478331801202831204088950080300194389999869} a^{2} + \frac{3167700257101767914519980420773233599839511552411011983975028039421103407791687823356355}{11130592977933274995111573614822137682187603848744875897429327531204956504315745589989389} a + \frac{331160589723523357200405735973883138375596585406087558040156979797316283719695902675}{3009081637721891050314023686083302968961233806094856960645938775670439714602796861311}$

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:  \( 132488202319000000000 \) (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

\(\Q(\sqrt{2701}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{37}, \sqrt{73})\), 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ 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$73.8.7.2$x^{8} - 1825$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.2$x^{8} - 1825$$8$$1$$7$$C_8$$[\ ]_{8}$