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Properties

Label 16.0.44664132968...161.25
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{8}$
Root discriminant $53.47$
Ramified primes $13, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10626213, -16161808, 15222337, -5978037, 1892735, 323596, -120419, -169455, 79004, -5012, 2477, -2325, 949, -109, 8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 8*x^14 - 109*x^13 + 949*x^12 - 2325*x^11 + 2477*x^10 - 5012*x^9 + 79004*x^8 - 169455*x^7 - 120419*x^6 + 323596*x^5 + 1892735*x^4 - 5978037*x^3 + 15222337*x^2 - 16161808*x + 10626213)
 
gp: K = bnfinit(x^16 - 6*x^15 + 8*x^14 - 109*x^13 + 949*x^12 - 2325*x^11 + 2477*x^10 - 5012*x^9 + 79004*x^8 - 169455*x^7 - 120419*x^6 + 323596*x^5 + 1892735*x^4 - 5978037*x^3 + 15222337*x^2 - 16161808*x + 10626213, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 8 x^{14} - 109 x^{13} + 949 x^{12} - 2325 x^{11} + 2477 x^{10} - 5012 x^{9} + 79004 x^{8} - 169455 x^{7} - 120419 x^{6} + 323596 x^{5} + 1892735 x^{4} - 5978037 x^{3} + 15222337 x^{2} - 16161808 x + 10626213 \)

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:  \(4466413296812760910104974161=13^{12}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.47$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{26} a^{12} - \frac{3}{13} a^{11} + \frac{4}{13} a^{10} + \frac{4}{13} a^{9} - \frac{11}{26} a^{7} + \frac{7}{26} a^{6} + \frac{3}{13} a^{5} + \frac{3}{26} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{78} a^{13} + \frac{1}{78} a^{12} - \frac{4}{39} a^{11} + \frac{2}{13} a^{10} + \frac{2}{39} a^{9} + \frac{5}{26} a^{8} - \frac{3}{13} a^{7} + \frac{1}{26} a^{6} - \frac{7}{78} a^{5} + \frac{4}{39} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{78} a^{14} - \frac{17}{39} a^{11} - \frac{7}{39} a^{10} + \frac{5}{78} a^{9} - \frac{11}{26} a^{8} - \frac{25}{78} a^{6} - \frac{3}{26} a^{5} + \frac{1}{13} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{84443744471985374313704479156152385465790532490066566} a^{15} + \frac{143788493686119497198287941321935174723406347584831}{84443744471985374313704479156152385465790532490066566} a^{14} - \frac{62470797927232442775632405620111259805124260885}{360870702871732368861984953658770878058933899530199} a^{13} - \frac{492398559309530098859903042760829615820872330067113}{84443744471985374313704479156152385465790532490066566} a^{12} - \frac{1220658219507496690185317571456488558473294404291901}{14073957411997562385617413192692064244298422081677761} a^{11} + \frac{4625853340064632076997182491239309658376383819961719}{28147914823995124771234826385384128488596844163355522} a^{10} + \frac{19954165498664799596857713157836175001795249035277734}{42221872235992687156852239578076192732895266245033283} a^{9} + \frac{398957124407262294092258670081991562696747819560147}{970617752551556026594304358116694085813684281495018} a^{8} - \frac{17838522939870610557617897918446061052900166257931037}{42221872235992687156852239578076192732895266245033283} a^{7} + \frac{40969377217603454428010609715286695517887912054200573}{84443744471985374313704479156152385465790532490066566} a^{6} - \frac{4458062660543710186270020453698381122085865673413689}{9382638274665041590411608795128042829532281387785174} a^{5} - \frac{634932417768397681001357103772819721553298382345425}{3247836325845591319757864582928937902530405095771791} a^{4} - \frac{173837404615648651964492978943462156988886146482197}{721741405743464737723969907317541756117867799060398} a^{3} - \frac{712706198517051882226883158040289032895240385438799}{2165224217230394213171909721952625268353603397181194} a^{2} + \frac{2425027813444776176137972686561233902256932969524985}{6495672651691182639515729165857875805060810191543582} a - \frac{182290902207616815538664729291931129898340090492977}{2165224217230394213171909721952625268353603397181194}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $7$
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:  \( 8818748.84847 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.294435349.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61Data not computed