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Properties

Label 16.0.63893255414...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}$
Root discriminant $199.68$
Ramified primes $5, 29, 89, 97$
Class number $29668416$ (GRH)
Class group $[2, 2, 4, 1854276]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7959188855021, -2276308091503, 2949128684464, -495918674948, 372148469851, -43319155511, 21895399022, -1741918829, 673687947, -33795292, 11307728, -313277, 106326, -1189, 521, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 521*x^14 - 1189*x^13 + 106326*x^12 - 313277*x^11 + 11307728*x^10 - 33795292*x^9 + 673687947*x^8 - 1741918829*x^7 + 21895399022*x^6 - 43319155511*x^5 + 372148469851*x^4 - 495918674948*x^3 + 2949128684464*x^2 - 2276308091503*x + 7959188855021)
 
gp: K = bnfinit(x^16 - x^15 + 521*x^14 - 1189*x^13 + 106326*x^12 - 313277*x^11 + 11307728*x^10 - 33795292*x^9 + 673687947*x^8 - 1741918829*x^7 + 21895399022*x^6 - 43319155511*x^5 + 372148469851*x^4 - 495918674948*x^3 + 2949128684464*x^2 - 2276308091503*x + 7959188855021, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 521 x^{14} - 1189 x^{13} + 106326 x^{12} - 313277 x^{11} + 11307728 x^{10} - 33795292 x^{9} + 673687947 x^{8} - 1741918829 x^{7} + 21895399022 x^{6} - 43319155511 x^{5} + 372148469851 x^{4} - 495918674948 x^{3} + 2949128684464 x^{2} - 2276308091503 x + 7959188855021 \)

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:  \(6389325541442866097681925088134765625=5^{12}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $199.68$
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:  $5, 29, 89, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} - \frac{9}{19} a^{13} + \frac{7}{19} a^{12} + \frac{1}{19} a^{11} - \frac{4}{19} a^{10} - \frac{8}{19} a^{9} + \frac{6}{19} a^{8} - \frac{7}{19} a^{7} + \frac{5}{19} a^{6} + \frac{8}{19} a^{5} - \frac{7}{19} a^{4} + \frac{6}{19}$, $\frac{1}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{15} + \frac{10704804783365228943414617807546311836295161011613898797815989039159640202053393704351294}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{14} + \frac{134662844048841260611137400365001390310889396898175181708232556089440323866510310537586127}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{13} - \frac{66174396256663250228096160385172176895923550889621801712199427541551224623632073602479125}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{12} - \frac{5794357254328978464132140795490187548255958863879816491633165988626112470082902497690210}{22864850090546070522193161614183896028557610528596297742377364917128230865866414561743261} a^{11} - \frac{62423279440980980833603824495977598205994944211651534068863740500862321689045530705747203}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{10} + \frac{137268825486803754735460521884576575771770607575434087182222156313108112608296905228456725}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{9} - \frac{131844036044092367010362158418481265931192603488606266841243970182336785523649292795656147}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{8} + \frac{182344095813089303394834041131079686183182483330051037778759625128277028989289652380837281}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{7} + \frac{212514224068245098819532323645234455015575173513628702890954474021189795957877761142968010}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{6} + \frac{169601537362163993301907036631488487709513397788155636686657767280313957022564325260881}{8196833051327836602295661710745170274388577359308106737833394970291252574555884465530603} a^{5} + \frac{98008846433298488993732543879683843552691952391436132094389453387176384830842203490103031}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a^{4} - \frac{1909062233790791227479545413815278017165729951554842485392690795281852664477371182081853}{22864850090546070522193161614183896028557610528596297742377364917128230865866414561743261} a^{3} - \frac{3746315814511294119598151344663144037320081115575834382225199327248499751837102388913277}{22864850090546070522193161614183896028557610528596297742377364917128230865866414561743261} a^{2} - \frac{214786018486905533442497407350804987640953970547704263371536090749816818770342718430082227}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959} a - \frac{98259902315387733563754677269015772288903162651332026738396489924680685920475615543411882}{434432151720375339921670070669494024542594600043329657105169933425436386451461876673121959}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{1854276}$, which has order $29668416$ (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:  $7$
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:  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:  \( 13998.0176198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.1, 4.4.725.1, 4.4.2225.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$