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Properties

Label 16.0.68969920686...1168.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 113^{3}\cdot 1009^{4}$
Root discriminant $200.64$
Ramified primes $2, 113, 1009$
Class number $24624704$ (GRH)
Class group $[2, 2, 2, 2, 1539044]$ (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12753433956482, -12251031213824, 6956085731328, -2392471589872, 631426271688, -136494175840, 28876699048, -3866300704, 761253712, -63644240, 11853904, -627488, 106852, -3440, 512, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 512*x^14 - 3440*x^13 + 106852*x^12 - 627488*x^11 + 11853904*x^10 - 63644240*x^9 + 761253712*x^8 - 3866300704*x^7 + 28876699048*x^6 - 136494175840*x^5 + 631426271688*x^4 - 2392471589872*x^3 + 6956085731328*x^2 - 12251031213824*x + 12753433956482)
 
gp: K = bnfinit(x^16 - 8*x^15 + 512*x^14 - 3440*x^13 + 106852*x^12 - 627488*x^11 + 11853904*x^10 - 63644240*x^9 + 761253712*x^8 - 3866300704*x^7 + 28876699048*x^6 - 136494175840*x^5 + 631426271688*x^4 - 2392471589872*x^3 + 6956085731328*x^2 - 12251031213824*x + 12753433956482, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 512 x^{14} - 3440 x^{13} + 106852 x^{12} - 627488 x^{11} + 11853904 x^{10} - 63644240 x^{9} + 761253712 x^{8} - 3866300704 x^{7} + 28876699048 x^{6} - 136494175840 x^{5} + 631426271688 x^{4} - 2392471589872 x^{3} + 6956085731328 x^{2} - 12251031213824 x + 12753433956482 \)

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:  \(6896992068684009105950840251291271168=2^{62}\cdot 113^{3}\cdot 1009^{4}\)
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 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}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{15} + \frac{116863693722481419975789066841117940175464790501935688544050457606771891205759525699650}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{14} + \frac{1258527458615798359996753384360260427385024337175121876861085127497446687941069177384651}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{13} + \frac{1774721959040688073204821826393886666963662389805898584066309739613640146192962259978154}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{12} + \frac{1617521898182957304325510773762149946120539311033714951203319847731724086423038246892685}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{11} - \frac{1045746480157908233145345254307283231185687934716781407358591885924596154777497697271079}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{10} - \frac{619176120539992385440344670582278854490828705397708049629568295427295131375477871626066}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{9} - \frac{754993309484139168222341682445667662841551484251143561108911143604192459645560887951015}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{8} - \frac{1306952632439970604199580693204583562536010580274130030044355194778361440807445751009254}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{7} + \frac{567705770887207912210989301028780938559394843596951479386694545740588152089547997045335}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{6} + \frac{902632990937903588670086976499154974197429312206176197434122654169167669142306109632105}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{5} - \frac{189941976918477413196381234510123014302020356381982606380101505735057246486051763874265}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{4} - \frac{1027837081859449773405786255506623228764472780092204057429866649115584405431685095909538}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{3} + \frac{495428559860963635114947771949947712803351102620577116761428826597345834903095914026716}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481} a^{2} + \frac{21507077899616959066577963445279814906728291965751253432451311580029382413423134485824}{219389194588925879469563539885647132923268203390098384644732721274910008458480373917793} a + \frac{1650788509731383229786688254439572280246719081467363052727354862258602780397245600456749}{3729616308011739950982580178056001259695559457631672538960456261673470143794166356602481}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1539044}$, which has order $24624704$ (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:  \( 32531.4048639 \) (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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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