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Properties

Label 16.16.6896992068...1168.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{62}\cdot 113^{3}\cdot 1009^{4}$
Root discriminant $200.64$
Ramified primes $2, 113, 1009$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![433872087934, -1808592898624, -995680684832, 614616142256, 365137419440, -29700370768, -32677192552, -369211456, 1164628660, 41048896, -18836896, -652752, 153884, 3888, -624, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 624*x^14 + 3888*x^13 + 153884*x^12 - 652752*x^11 - 18836896*x^10 + 41048896*x^9 + 1164628660*x^8 - 369211456*x^7 - 32677192552*x^6 - 29700370768*x^5 + 365137419440*x^4 + 614616142256*x^3 - 995680684832*x^2 - 1808592898624*x + 433872087934)
 
gp: K = bnfinit(x^16 - 8*x^15 - 624*x^14 + 3888*x^13 + 153884*x^12 - 652752*x^11 - 18836896*x^10 + 41048896*x^9 + 1164628660*x^8 - 369211456*x^7 - 32677192552*x^6 - 29700370768*x^5 + 365137419440*x^4 + 614616142256*x^3 - 995680684832*x^2 - 1808592898624*x + 433872087934, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 624 x^{14} + 3888 x^{13} + 153884 x^{12} - 652752 x^{11} - 18836896 x^{10} + 41048896 x^{9} + 1164628660 x^{8} - 369211456 x^{7} - 32677192552 x^{6} - 29700370768 x^{5} + 365137419440 x^{4} + 614616142256 x^{3} - 995680684832 x^{2} - 1808592898624 x + 433872087934 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6896992068684009105950840251291271168=2^{62}\cdot 113^{3}\cdot 1009^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.64$
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:  $2, 113, 1009$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{15} - \frac{342425812559226406258129885272848056754915811641467347186010204824252139875107426394273}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{14} + \frac{1366918231621701574821679379043879945474779421219424616732634400850884184514127014587539}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{13} + \frac{577691171786113880230089042102233998163472923266192537879463819961271463861708012070820}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{12} + \frac{1404448556774316436177644599923394370369141851967658529534384332350134375439084855883266}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{11} - \frac{1394338994609840028882316104600878853464125658580459489015075402822101267621173403773072}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{10} + \frac{1550302428337747111476215742892514940039897760757648880749320374656357109688756966229399}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{9} + \frac{1842738096272694341883107125370562887828497577218252190025264720707533517322378521445802}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{8} - \frac{356994733861678490399375560738722702763206051353651296061096535970740441673545777680116}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{7} + \frac{533921210441010159254099385663751101711920329862036266830879291992148943434764136429933}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{6} + \frac{1901709901171076319848238873473693307928331596812727653327357530820513056280395138668648}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{5} + \frac{308913843376076331027780391146968864611840701124708650756956112106658815682303851140199}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{4} - \frac{1507247153359412211225671263297533015955217519526257802776297622222755837379808777814053}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{3} + \frac{1280988364153681228990789969521438795073553000436636137796543648205443842371088875939944}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a^{2} - \frac{879347485970157036080344242198572324830479517480074727173488101420908741288510380641667}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217} a + \frac{1966571567640507247894989219305615986812597526268126627357590231342657744503752483214593}{5241266853003378888031306964556931674040108308912877088821588565834772163680532034709217}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 26390203117300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7583301632.1

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 R $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
$113$113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
1009Data not computed