Properties

Label 16.0.876...713.5
Degree $16$
Signature $[0, 8]$
Discriminant $8.768\times 10^{36}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $253504$ (GRH)
Class group $[2, 2, 4, 15844]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 177*x^14 - 652*x^13 + 12492*x^12 - 48379*x^11 + 504319*x^10 - 2195433*x^9 + 13672084*x^8 - 78136205*x^7 + 244603409*x^6 - 998379714*x^5 + 4521909355*x^4 - 12093973889*x^3 + 20305914608*x^2 - 24281793064*x + 15386186173)
 
gp: K = bnfinit(x^16 - 3*x^15 + 177*x^14 - 652*x^13 + 12492*x^12 - 48379*x^11 + 504319*x^10 - 2195433*x^9 + 13672084*x^8 - 78136205*x^7 + 244603409*x^6 - 998379714*x^5 + 4521909355*x^4 - 12093973889*x^3 + 20305914608*x^2 - 24281793064*x + 15386186173, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15386186173, -24281793064, 20305914608, -12093973889, 4521909355, -998379714, 244603409, -78136205, 13672084, -2195433, 504319, -48379, 12492, -652, 177, -3, 1]);
 

\( x^{16} - 3 x^{15} + 177 x^{14} - 652 x^{13} + 12492 x^{12} - 48379 x^{11} + 504319 x^{10} - 2195433 x^{9} + 13672084 x^{8} - 78136205 x^{7} + 244603409 x^{6} - 998379714 x^{5} + 4521909355 x^{4} - 12093973889 x^{3} + 20305914608 x^{2} - 24281793064 x + 15386186173 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(8767895277848913089642817986417897713\)\(\medspace = 61^{4}\cdot 97^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $203.67$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $61, 97$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{206} a^{13} + \frac{23}{206} a^{12} + \frac{42}{103} a^{11} - \frac{95}{206} a^{10} + \frac{39}{206} a^{9} - \frac{6}{103} a^{8} - \frac{17}{103} a^{7} - \frac{14}{103} a^{6} - \frac{18}{103} a^{5} - \frac{35}{206} a^{4} + \frac{95}{206} a^{3} - \frac{33}{206} a^{2} - \frac{39}{103} a + \frac{45}{206}$, $\frac{1}{206} a^{14} - \frac{33}{206} a^{12} + \frac{33}{206} a^{11} - \frac{21}{103} a^{10} - \frac{85}{206} a^{9} + \frac{18}{103} a^{8} - \frac{35}{103} a^{7} - \frac{5}{103} a^{6} - \frac{31}{206} a^{5} + \frac{38}{103} a^{4} + \frac{24}{103} a^{3} + \frac{63}{206} a^{2} - \frac{15}{206} a - \frac{5}{206}$, $\frac{1}{336228505572068100697504858693061614410864283868943686635733135407157278} a^{15} + \frac{125858481541907834433719675404337685306709249973411405954535243584331}{336228505572068100697504858693061614410864283868943686635733135407157278} a^{14} - \frac{194381248566220758583718155807568396635935168977381987269128258052865}{336228505572068100697504858693061614410864283868943686635733135407157278} a^{13} - \frac{31012942346398309921882818127224850821868853994380907862162640247016612}{168114252786034050348752429346530807205432141934471843317866567703578639} a^{12} + \frac{153304867815580155934502529884059179520710255258658394963446025114940269}{336228505572068100697504858693061614410864283868943686635733135407157278} a^{11} - \frac{78449503145398789324864065830697233253864538185949767727262714355245725}{336228505572068100697504858693061614410864283868943686635733135407157278} a^{10} + \frac{123110676296211085129583487285678586830258285903543800936743514733320317}{336228505572068100697504858693061614410864283868943686635733135407157278} a^{9} - \frac{17713966102003803756026505912901127818962806835116621342144520676750010}{168114252786034050348752429346530807205432141934471843317866567703578639} a^{8} + \frac{145669531481300377436673359598452858948529384681652741543152953920599}{1632177211514893692706334265500299099081865455674483915707442404889113} a^{7} + \frac{22546015104109509682964961235632206642777081044784615709918033893250861}{336228505572068100697504858693061614410864283868943686635733135407157278} a^{6} - \frac{60711830132122356007953112866391234117941545213735360418110995806830521}{336228505572068100697504858693061614410864283868943686635733135407157278} a^{5} + \frac{47166149218019358697244725677428054496679997173789125375280838994405858}{168114252786034050348752429346530807205432141934471843317866567703578639} a^{4} - \frac{61109719720819480017558727638104146559480041226263039469267221673013261}{336228505572068100697504858693061614410864283868943686635733135407157278} a^{3} - \frac{28242201650598369013210667010658190075560863508325523224713710262044114}{168114252786034050348752429346530807205432141934471843317866567703578639} a^{2} + \frac{7880399046434316867295627023647076327187021931814952704257528401139894}{168114252786034050348752429346530807205432141934471843317866567703578639} a - \frac{153622141904299432521140178192789627003329051195548282799403660404771783}{336228505572068100697504858693061614410864283868943686635733135407157278}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{15844}$, which has order $253504$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1675810.87182 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 1675810.87182 \cdot 253504}{2\sqrt{8767895277848913089642817986417897713}}\approx 0.174249320879$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

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

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
97Data not computed