↑

Properties

Label 16.0.13279393802...0625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{11}$
Root discriminant $76.33$
Ramified primes $5, 61$
Class number $32$ (GRH)
Class group $[2, 4, 4]$ (GRH)
Galois group 16T1192

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![422035031, -232326241, 190084305, -131725735, -147940, 14725862, 5748138, -4353125, -42645, 404020, -44417, -17008, 3330, 320, -90, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 90*x^14 + 320*x^13 + 3330*x^12 - 17008*x^11 - 44417*x^10 + 404020*x^9 - 42645*x^8 - 4353125*x^7 + 5748138*x^6 + 14725862*x^5 - 147940*x^4 - 131725735*x^3 + 190084305*x^2 - 232326241*x + 422035031)
 
gp: K = bnfinit(x^16 - x^15 - 90*x^14 + 320*x^13 + 3330*x^12 - 17008*x^11 - 44417*x^10 + 404020*x^9 - 42645*x^8 - 4353125*x^7 + 5748138*x^6 + 14725862*x^5 - 147940*x^4 - 131725735*x^3 + 190084305*x^2 - 232326241*x + 422035031, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 90 x^{14} + 320 x^{13} + 3330 x^{12} - 17008 x^{11} - 44417 x^{10} + 404020 x^{9} - 42645 x^{8} - 4353125 x^{7} + 5748138 x^{6} + 14725862 x^{5} - 147940 x^{4} - 131725735 x^{3} + 190084305 x^{2} - 232326241 x + 422035031 \)

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:  \(1327939380231806599761962890625=5^{15}\cdot 61^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.33$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{13} - \frac{4}{11} a^{12} + \frac{3}{11} a^{11} + \frac{5}{11} a^{10} + \frac{3}{11} a^{9} + \frac{2}{11} a^{8} - \frac{5}{11} a^{7} - \frac{2}{11} a^{6} + \frac{3}{11} a^{5} + \frac{4}{11} a^{4} + \frac{2}{11} a^{3} + \frac{2}{11} a^{2} - \frac{2}{11} a$, $\frac{1}{6019528658459058324838381852746347102160862474323071233277931701} a^{15} + \frac{39142019137016904817203871164616964157094419870020446461218985}{6019528658459058324838381852746347102160862474323071233277931701} a^{14} + \frac{2806249903304061934445525160442339070467171251999516066733592182}{6019528658459058324838381852746347102160862474323071233277931701} a^{13} + \frac{1433755957488397225825609086989195827952494774590548204869758617}{6019528658459058324838381852746347102160862474323071233277931701} a^{12} - \frac{650758027825453272503668570063092836969717572719616825130919408}{6019528658459058324838381852746347102160862474323071233277931701} a^{11} - \frac{427284739648333020180542123133541759436567906085485574599677498}{6019528658459058324838381852746347102160862474323071233277931701} a^{10} - \frac{778137380100546320565494221821327082892687342464406268899843122}{6019528658459058324838381852746347102160862474323071233277931701} a^{9} - \frac{2489456173517912300955317631687618275345593082305993354709803404}{6019528658459058324838381852746347102160862474323071233277931701} a^{8} + \frac{1445172143986532117785577290328381677223956352292514129944371}{3023369491943273894946449951153363687674968595842828344187811} a^{7} + \frac{1351537395702367758170782015101210022293758527148609524886589335}{6019528658459058324838381852746347102160862474323071233277931701} a^{6} - \frac{2448650688924732798056876698965384961740803775410635333315813937}{6019528658459058324838381852746347102160862474323071233277931701} a^{5} + \frac{1286990807258274022524596831121636769601673916344647987670451182}{6019528658459058324838381852746347102160862474323071233277931701} a^{4} - \frac{1837972341029101293262152029437016368664427413345643155429930617}{6019528658459058324838381852746347102160862474323071233277931701} a^{3} + \frac{1561725027021760742754183553830574703614432103666535810155059333}{6019528658459058324838381852746347102160862474323071233277931701} a^{2} + \frac{2106514136928067727799063507861608971539333190818675943464558562}{6019528658459058324838381852746347102160862474323071233277931701} a + \frac{102273280544468976195541464500409552970616112188341588753824850}{547229878041732574985307441158758827469169315847551930297993791}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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:  \( -\frac{98480885696949138906253414987530292940651461606687}{3627662221604274422700567184651224252375408354475102605961} a^{15} - \frac{183912588965479878348041685528189874407994499063628}{3627662221604274422700567184651224252375408354475102605961} a^{14} + \frac{8379286576198283063490280495238523547034946675523917}{3627662221604274422700567184651224252375408354475102605961} a^{13} - \frac{7306180706205745607988909990737672434706619293307112}{3627662221604274422700567184651224252375408354475102605961} a^{12} - \frac{352216294365426875596990024830092068696631364337367378}{3627662221604274422700567184651224252375408354475102605961} a^{11} + \frac{660909745659486054417720694668544214318845631233520203}{3627662221604274422700567184651224252375408354475102605961} a^{10} + \frac{6413610238032073296899271664864846371244319159533860505}{3627662221604274422700567184651224252375408354475102605961} a^{9} - \frac{1941921716899984587615494568897631072191310975127070800}{329787474691297674790960653150111295670491668588645691451} a^{8} - \frac{329333432926381624776362638639830977920644371737862797}{20042332716045715042544570080945990344615515770580677381} a^{7} + \frac{259950524455734066751819605230436529284243545081621655669}{3627662221604274422700567184651224252375408354475102605961} a^{6} + \frac{209967863487454190548295675280793346028134151489653719448}{3627662221604274422700567184651224252375408354475102605961} a^{5} - \frac{854496394337806114273936889301026470897070375731196431151}{3627662221604274422700567184651224252375408354475102605961} a^{4} - \frac{2689500844031862968860769175716948464825841708727956305386}{3627662221604274422700567184651224252375408354475102605961} a^{3} + \frac{4969829298879850591643808811619095163677506816945654545000}{3627662221604274422700567184651224252375408354475102605961} a^{2} - \frac{1640450758576175520893021769207163457983253706734827861485}{3627662221604274422700567184651224252375408354475102605961} a + \frac{1505231813640493929737213136851408096863661375850978758006}{329787474691297674790960653150111295670491668588645691451} \) (order $10$)
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:  \( 98083548.7284 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.17732890625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$