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Properties

Label 16.8.34487004091...1249.1
Degree $16$
Signature $[8, 4]$
Discriminant $41^{4}\cdot 73^{14}$
Root discriminant $108.04$
Ramified primes $41, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![245954816, -42093568, -178366000, 28122920, 49136920, -7089206, -6975475, 856419, 628982, -54891, -42649, 1730, 2135, -3, -66, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 66*x^14 - 3*x^13 + 2135*x^12 + 1730*x^11 - 42649*x^10 - 54891*x^9 + 628982*x^8 + 856419*x^7 - 6975475*x^6 - 7089206*x^5 + 49136920*x^4 + 28122920*x^3 - 178366000*x^2 - 42093568*x + 245954816)
 
gp: K = bnfinit(x^16 - x^15 - 66*x^14 - 3*x^13 + 2135*x^12 + 1730*x^11 - 42649*x^10 - 54891*x^9 + 628982*x^8 + 856419*x^7 - 6975475*x^6 - 7089206*x^5 + 49136920*x^4 + 28122920*x^3 - 178366000*x^2 - 42093568*x + 245954816, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 66 x^{14} - 3 x^{13} + 2135 x^{12} + 1730 x^{11} - 42649 x^{10} - 54891 x^{9} + 628982 x^{8} + 856419 x^{7} - 6975475 x^{6} - 7089206 x^{5} + 49136920 x^{4} + 28122920 x^{3} - 178366000 x^{2} - 42093568 x + 245954816 \)

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:  \(344870040917968360157170212631249=41^{4}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.04$
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:  $41, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} - \frac{3}{16} a^{4}$, $\frac{1}{96} a^{12} - \frac{1}{48} a^{11} - \frac{5}{96} a^{10} - \frac{1}{8} a^{9} - \frac{5}{48} a^{8} - \frac{1}{48} a^{7} - \frac{17}{96} a^{6} - \frac{1}{24} a^{5} + \frac{23}{96} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{96} a^{13} - \frac{1}{32} a^{11} - \frac{1}{24} a^{10} + \frac{1}{48} a^{9} - \frac{5}{48} a^{8} + \frac{1}{32} a^{7} + \frac{11}{48} a^{6} + \frac{7}{32} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{96} a^{14} + \frac{1}{48} a^{11} - \frac{1}{96} a^{10} + \frac{1}{48} a^{9} - \frac{1}{32} a^{8} + \frac{1}{6} a^{7} - \frac{1}{16} a^{6} + \frac{1}{6} a^{5} + \frac{3}{32} a^{4} - \frac{3}{8} a^{3} - \frac{5}{12} a^{2} - \frac{1}{12} a$, $\frac{1}{134704227350182306455646881779102508103535616} a^{15} + \frac{132371009088242115733127167019986492868161}{44901409116727435485215627259700836034511872} a^{14} + \frac{207491515119651770528291575318875261752677}{67352113675091153227823440889551254051767808} a^{13} - \frac{617713729300104983379101864914549195797851}{134704227350182306455646881779102508103535616} a^{12} - \frac{3411584745158012880226528481708146112356949}{134704227350182306455646881779102508103535616} a^{11} + \frac{224604931723889123005036918735611463150903}{67352113675091153227823440889551254051767808} a^{10} + \frac{48148257459341799429550284090447248954039}{1513530644384070859052212154821376495545344} a^{9} - \frac{2926259125145335990489840898790542032853477}{44901409116727435485215627259700836034511872} a^{8} - \frac{12743491809918868988037485743878636532608611}{67352113675091153227823440889551254051767808} a^{7} + \frac{31051013152249854199197314851571159846010059}{134704227350182306455646881779102508103535616} a^{6} - \frac{28858027048962338515669082001836440569473927}{134704227350182306455646881779102508103535616} a^{5} + \frac{3464729913510602007759002906233070483966109}{22450704558363717742607813629850418017255936} a^{4} + \frac{1521318428693900614102783813528117846231829}{8419014209386394153477930111193906756470976} a^{3} + \frac{6515839123311960578910393541662567308019245}{16838028418772788306955860222387813512941952} a^{2} - \frac{1513693750510970387203702721774558665703657}{8419014209386394153477930111193906756470976} a + \frac{349186419230557964901483366400116406505629}{701584517448866179456494175932825563039248}$

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:  $11$
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:  \( 158755699378 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.18570676910602057.1, 8.4.452943339282977.1, 8.4.6204703277849.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
41Data not computed
73Data not computed