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Properties

Label 16.8.90390672967...1729.5
Degree $16$
Signature $[8, 4]$
Discriminant $61^{4}\cdot 97^{14}$
Root discriminant $153.02$
Ramified primes $61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4289443, 21707769, -16279720, -15094129, 3175065, 2803743, 112567, 843227, -223672, -117429, -1111, 7006, 1228, -88, -53, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 53*x^14 - 88*x^13 + 1228*x^12 + 7006*x^11 - 1111*x^10 - 117429*x^9 - 223672*x^8 + 843227*x^7 + 112567*x^6 + 2803743*x^5 + 3175065*x^4 - 15094129*x^3 - 16279720*x^2 + 21707769*x + 4289443)
 
gp: K = bnfinit(x^16 - 4*x^15 - 53*x^14 - 88*x^13 + 1228*x^12 + 7006*x^11 - 1111*x^10 - 117429*x^9 - 223672*x^8 + 843227*x^7 + 112567*x^6 + 2803743*x^5 + 3175065*x^4 - 15094129*x^3 - 16279720*x^2 + 21707769*x + 4289443, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 53 x^{14} - 88 x^{13} + 1228 x^{12} + 7006 x^{11} - 1111 x^{10} - 117429 x^{9} - 223672 x^{8} + 843227 x^{7} + 112567 x^{6} + 2803743 x^{5} + 3175065 x^{4} - 15094129 x^{3} - 16279720 x^{2} + 21707769 x + 4289443 \)

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:  \(90390672967514567934462041097091729=61^{4}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.02$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6}$, $\frac{1}{24} a^{12} + \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} + \frac{1}{12} a^{6} - \frac{7}{24} a^{5} - \frac{1}{4} a^{4} + \frac{11}{24} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a + \frac{1}{24}$, $\frac{1}{24} a^{13} - \frac{1}{12} a^{11} - \frac{5}{24} a^{9} + \frac{1}{8} a^{8} - \frac{7}{24} a^{7} - \frac{3}{8} a^{6} + \frac{1}{24} a^{5} + \frac{5}{24} a^{4} - \frac{1}{3} a^{3} + \frac{1}{8} a^{2} - \frac{5}{24} a + \frac{11}{24}$, $\frac{1}{792} a^{14} - \frac{1}{396} a^{13} + \frac{5}{396} a^{12} - \frac{1}{198} a^{11} - \frac{19}{264} a^{10} - \frac{25}{264} a^{9} + \frac{167}{792} a^{8} + \frac{277}{792} a^{7} + \frac{65}{264} a^{6} + \frac{61}{264} a^{5} - \frac{97}{396} a^{4} + \frac{287}{792} a^{3} + \frac{185}{792} a^{2} + \frac{101}{792} a + \frac{97}{396}$, $\frac{1}{2447574791810849139108159309936203160887155830287976} a^{15} + \frac{53862463736760845833021865360836742171613157703}{222506799255531739918923573630563923717014166389816} a^{14} + \frac{37479315382086199808313092428942779648943045454515}{2447574791810849139108159309936203160887155830287976} a^{13} + \frac{24153996142302030315136975787989241654842677495223}{1223787395905424569554079654968101580443577915143988} a^{12} - \frac{13828831756844387777086314602905309651835516045191}{271952754645649904345351034437355906765239536698664} a^{11} - \frac{21140180976397452349777414140930882843169678214503}{407929131968474856518026551656033860147859305047996} a^{10} + \frac{473991752298936143254595018356305744392626151357087}{2447574791810849139108159309936203160887155830287976} a^{9} + \frac{18863836684380638779719138030417955191687008770231}{2447574791810849139108159309936203160887155830287976} a^{8} + \frac{231589803582457223703998292902554441547058458440379}{815858263936949713036053103312067720295718610095992} a^{7} + \frac{1839708150456952148292970930198706892136679995953}{271952754645649904345351034437355906765239536698664} a^{6} + \frac{332871768764073346946056794404571007055427065917703}{1223787395905424569554079654968101580443577915143988} a^{5} + \frac{220767990485339716469404590592957905045678396469025}{611893697952712284777039827484050790221788957571994} a^{4} - \frac{33119389320905554095719489737114177393307554141879}{111253399627765869959461786815281961858507083194908} a^{3} + \frac{292318782926574921758419192053329776894222853121521}{2447574791810849139108159309936203160887155830287976} a^{2} - \frac{28761918895630300988646126986385128392280280749941}{111253399627765869959461786815281961858507083194908} a - \frac{135742764853703659981927959531125680782199846708077}{271952754645649904345351034437355906765239536698664}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 164841310865 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

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 t16n817
Character table for t16n817 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$