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Properties

Label 16.8.87678952778...7713.2
Degree $16$
Signature $[8, 4]$
Discriminant $61^{4}\cdot 97^{15}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-939737, -8408452, 16830832, 15105765, -51369568, 36806443, -5743921, -3231766, 1291073, -314877, 25855, 8768, -8134, 125, 39, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 39*x^14 + 125*x^13 - 8134*x^12 + 8768*x^11 + 25855*x^10 - 314877*x^9 + 1291073*x^8 - 3231766*x^7 - 5743921*x^6 + 36806443*x^5 - 51369568*x^4 + 15105765*x^3 + 16830832*x^2 - 8408452*x - 939737)
 
gp: K = bnfinit(x^16 - 5*x^15 + 39*x^14 + 125*x^13 - 8134*x^12 + 8768*x^11 + 25855*x^10 - 314877*x^9 + 1291073*x^8 - 3231766*x^7 - 5743921*x^6 + 36806443*x^5 - 51369568*x^4 + 15105765*x^3 + 16830832*x^2 - 8408452*x - 939737, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 39 x^{14} + 125 x^{13} - 8134 x^{12} + 8768 x^{11} + 25855 x^{10} - 314877 x^{9} + 1291073 x^{8} - 3231766 x^{7} - 5743921 x^{6} + 36806443 x^{5} - 51369568 x^{4} + 15105765 x^{3} + 16830832 x^{2} - 8408452 x - 939737 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8767895277848913089642817986417897713=61^{4}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.67$
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:  $61, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{14} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} + \frac{3}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{3540693326243118528497534230117929654037818276245425708923448} a^{15} + \frac{66010667535231425269826442152673396057986783394297513176217}{1770346663121559264248767115058964827018909138122712854461724} a^{14} + \frac{11908322848028166809867814217202825836092079007306672417843}{442586665780389816062191778764741206754727284530678213615431} a^{13} + \frac{270562869902981003714221655512091059491519966912828731208121}{3540693326243118528497534230117929654037818276245425708923448} a^{12} - \frac{245647556291748342139914957695364865826886163271844222503179}{3540693326243118528497534230117929654037818276245425708923448} a^{11} + \frac{342675290697596072772446509999631528672246604888160442174217}{1770346663121559264248767115058964827018909138122712854461724} a^{10} - \frac{294875799210440164231558451867798206191033028674443614357275}{1770346663121559264248767115058964827018909138122712854461724} a^{9} - \frac{1170607988888141380660666273023569197509882034514430764105023}{3540693326243118528497534230117929654037818276245425708923448} a^{8} - \frac{600202264554842764729639410255453617636048872296825560161423}{1770346663121559264248767115058964827018909138122712854461724} a^{7} - \frac{396935980629284608779994067331833324848743470293009768339261}{3540693326243118528497534230117929654037818276245425708923448} a^{6} + \frac{143229506315076082377055266358770652775192709956115604021742}{442586665780389816062191778764741206754727284530678213615431} a^{5} - \frac{252531843047513306348533446484717552648532010076571152115681}{885173331560779632124383557529482413509454569061356427230862} a^{4} + \frac{274339272582658243582853295194061431773480712502051915018155}{1770346663121559264248767115058964827018909138122712854461724} a^{3} - \frac{561432455400618624449401045441200426409310755535378261888189}{3540693326243118528497534230117929654037818276245425708923448} a^{2} - \frac{528619389190955287940321671695016998147585940435188338472085}{3540693326243118528497534230117929654037818276245425708923448} a + \frac{129385911678809136225334919886468568879008861846756579542935}{1770346663121559264248767115058964827018909138122712854461724}$

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

Class group and class number

$C_{4}$, 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:  $11$
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:  \( 625132198351 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
61Data not computed
97Data not computed