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Properties

Label 16.0.33662328752...4641.9
Degree $16$
Signature $[0, 8]$
Discriminant $31^{6}\cdot 41^{14}$
Root discriminant $93.42$
Ramified primes $31, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![6958823, -14658053, 18066458, -7546349, 5675872, -2505376, 1209077, -326850, 87100, 295, -3898, 2016, -245, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 245*x^12 + 2016*x^11 - 3898*x^10 + 295*x^9 + 87100*x^8 - 326850*x^7 + 1209077*x^6 - 2505376*x^5 + 5675872*x^4 - 7546349*x^3 + 18066458*x^2 - 14658053*x + 6958823)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 245*x^12 + 2016*x^11 - 3898*x^10 + 295*x^9 + 87100*x^8 - 326850*x^7 + 1209077*x^6 - 2505376*x^5 + 5675872*x^4 - 7546349*x^3 + 18066458*x^2 - 14658053*x + 6958823, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 245 x^{12} + 2016 x^{11} - 3898 x^{10} + 295 x^{9} + 87100 x^{8} - 326850 x^{7} + 1209077 x^{6} - 2505376 x^{5} + 5675872 x^{4} - 7546349 x^{3} + 18066458 x^{2} - 14658053 x + 6958823 \)

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:  \(33662328752972862921089838254641=31^{6}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.42$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{41} a^{8} - \frac{4}{41} a^{7} + \frac{7}{41} a^{6} - \frac{7}{41} a^{5} - \frac{11}{41} a^{4} - \frac{12}{41} a^{3} + \frac{3}{41} a^{2} - \frac{18}{41} a - \frac{4}{41}$, $\frac{1}{41} a^{9} - \frac{9}{41} a^{7} - \frac{20}{41} a^{6} + \frac{2}{41} a^{5} - \frac{15}{41} a^{4} - \frac{4}{41} a^{3} - \frac{6}{41} a^{2} + \frac{6}{41} a - \frac{16}{41}$, $\frac{1}{41} a^{10} - \frac{15}{41} a^{7} - \frac{17}{41} a^{6} + \frac{4}{41} a^{5} + \frac{20}{41} a^{4} + \frac{9}{41} a^{3} - \frac{8}{41} a^{2} - \frac{14}{41} a + \frac{5}{41}$, $\frac{1}{41} a^{11} + \frac{5}{41} a^{7} - \frac{14}{41} a^{6} - \frac{3}{41} a^{5} + \frac{8}{41} a^{4} + \frac{17}{41} a^{3} - \frac{10}{41} a^{2} - \frac{19}{41} a - \frac{19}{41}$, $\frac{1}{82} a^{12} - \frac{1}{82} a^{9} - \frac{13}{41} a^{7} + \frac{23}{82} a^{6} - \frac{18}{41} a^{4} + \frac{13}{82} a^{3} - \frac{14}{41} a^{2} + \frac{12}{41} a - \frac{5}{82}$, $\frac{1}{410} a^{13} + \frac{1}{410} a^{12} + \frac{1}{205} a^{11} + \frac{3}{410} a^{10} - \frac{1}{82} a^{9} + \frac{2}{205} a^{8} + \frac{109}{410} a^{7} + \frac{53}{410} a^{6} + \frac{1}{205} a^{5} - \frac{23}{82} a^{4} + \frac{39}{410} a^{3} - \frac{53}{205} a^{2} - \frac{13}{82} a - \frac{79}{410}$, $\frac{1}{297694925010648391550} a^{14} - \frac{7}{297694925010648391550} a^{13} - \frac{1625744571590559371}{297694925010648391550} a^{12} + \frac{2493615600015346767}{297694925010648391550} a^{11} + \frac{2723690445998672071}{297694925010648391550} a^{10} - \frac{1382478054081993061}{297694925010648391550} a^{9} + \frac{3242137695577650667}{297694925010648391550} a^{8} + \frac{59060348289990218171}{297694925010648391550} a^{7} - \frac{47548243757383476877}{297694925010648391550} a^{6} - \frac{124893904122334954851}{297694925010648391550} a^{5} + \frac{23384917781792047789}{297694925010648391550} a^{4} - \frac{83598505356646623993}{297694925010648391550} a^{3} + \frac{14471531934553915163}{297694925010648391550} a^{2} - \frac{144022290896538634019}{297694925010648391550} a - \frac{6246155498568226883}{297694925010648391550}$, $\frac{1}{695117649899863994269250} a^{15} + \frac{116}{69511764989986399426925} a^{14} + \frac{109313713711554830697}{139023529979972798853850} a^{13} - \frac{121223022747053454149}{69511764989986399426925} a^{12} + \frac{820593407938982731111}{69511764989986399426925} a^{11} + \frac{4662008471564846266771}{695117649899863994269250} a^{10} - \frac{537123099911361719842}{69511764989986399426925} a^{9} + \frac{817736358125733024791}{69511764989986399426925} a^{8} - \frac{67166622297897151834799}{139023529979972798853850} a^{7} + \frac{24672817581255746519169}{69511764989986399426925} a^{6} - \frac{119910590025918288010889}{347558824949931997134625} a^{5} + \frac{69476950649355309253469}{139023529979972798853850} a^{4} - \frac{146554383697889732752184}{347558824949931997134625} a^{3} + \frac{21906355548604149981801}{347558824949931997134625} a^{2} + \frac{104437661660000864308319}{695117649899863994269250} a - \frac{32128637575644441443011}{695117649899863994269250}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $7$
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:  \( 294383359.09 \) (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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.4.187158857199641.1, 8.2.6037382490311.1, 8.2.141510355443631.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
31Data not computed
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$