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Properties

Label 16.6.55013835857...3056.4
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{48}\cdot 3^{11}\cdot 7^{8}\cdot 294337^{7}$
Root discriminant $11{,}124.47$
Ramified primes $2, 3, 7, 294337$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-402460802329301763291, 0, -272538091023140842308, 0, 477521499921995448, 0, 16943428675424988, 0, -64623851152335, 0, 51851891316, 0, -7793522, 0, -4064, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4064*x^14 - 7793522*x^12 + 51851891316*x^10 - 64623851152335*x^8 + 16943428675424988*x^6 + 477521499921995448*x^4 - 272538091023140842308*x^2 - 402460802329301763291)
 
gp: K = bnfinit(x^16 - 4064*x^14 - 7793522*x^12 + 51851891316*x^10 - 64623851152335*x^8 + 16943428675424988*x^6 + 477521499921995448*x^4 - 272538091023140842308*x^2 - 402460802329301763291, 1)
 

Normalized defining polynomial

\( x^{16} - 4064 x^{14} - 7793522 x^{12} + 51851891316 x^{10} - 64623851152335 x^{8} + 16943428675424988 x^{6} + 477521499921995448 x^{4} - 272538091023140842308 x^{2} - 402460802329301763291 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-55013835857676939099088270698740000784693073140191887871554093056=-\,2^{48}\cdot 3^{11}\cdot 7^{8}\cdot 294337^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}124.47$
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:  $2, 3, 7, 294337$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{45} a^{12} - \frac{4}{45} a^{8} - \frac{4}{15} a^{4} - \frac{2}{5}$, $\frac{1}{494955} a^{13} + \frac{5269}{98991} a^{11} - \frac{4289}{494955} a^{9} - \frac{13100}{98991} a^{7} + \frac{5487}{54995} a^{5} - \frac{12799}{32997} a^{3} + \frac{20413}{54995} a$, $\frac{1}{219933028740653593092501458328089942756787605016482672259527796243188595} a^{14} - \frac{2108662968714583089003248926697618405691013860858294734974095315708044}{219933028740653593092501458328089942756787605016482672259527796243188595} a^{12} + \frac{11057791092376275095071820076301911536828288619517166161047953593847881}{219933028740653593092501458328089942756787605016482672259527796243188595} a^{10} + \frac{8039652554812748242563885510673644968857390271076542237804922537972377}{73311009580217864364167152776029980918929201672160890753175932081062865} a^{8} - \frac{6894403873364584350101702294248132031117544585225803543194373492335004}{73311009580217864364167152776029980918929201672160890753175932081062865} a^{6} + \frac{8755748150763911181826503002497259512935689863367774272469603436178477}{24437003193405954788055717592009993639643067224053630251058644027020955} a^{4} - \frac{12216299132453116937081238907936015333529764179213004936036956921964397}{24437003193405954788055717592009993639643067224053630251058644027020955} a^{2} - \frac{145588823625255618758933495937416206925474493462496721535904323}{1144640763483109714018576537355112168627341712065793169200508745}$, $\frac{1}{142296669595202874730848443538274192963641580445664288951914484169343020965} a^{15} - \frac{96203882198798577045719242649736576543937736524466361357501101718083}{142296669595202874730848443538274192963641580445664288951914484169343020965} a^{13} - \frac{219874470172430219413955330583615677910252728075432788477588679624135767}{8370392329129580866520496678722011350802445908568487585406734362902530645} a^{11} + \frac{2441869339232973664918058237213723874247347501209468885320692696476212354}{47432223198400958243616147846091397654547193481888096317304828056447673655} a^{9} + \frac{512304542869959896493764190887132143729092034998602001631261959564906009}{5270247022044550915957349760676821961616354831320899590811647561827519295} a^{7} + \frac{2076294486343588372261043424849973148162236817289944785810383211422319788}{5270247022044550915957349760676821961616354831320899590811647561827519295} a^{5} - \frac{4062050632282788996714471926901435353417122002957669532033049884694143057}{15810741066133652747872049282030465884849064493962698772434942685482557885} a^{3} + \frac{2411801288336945663289079763158466510551056072700308508211602671463}{12589903757550723744490323334368878742732131491011659068036395686255} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  $10$
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:  \( 76246431647900000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n1022
Character table for t16n1022 is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), 4.4.2076841872.1, 8.6.975018691690245188654137344.1

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
Arithmetically equvalently 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
294337Data not computed