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Properties

Label 16.0.29630251663...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $33.84$
Ramified primes $2, 5, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16631, 31856, 65772, 93808, 98107, 83536, 52532, 18704, 1550, -1084, -386, -112, -28, -8, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 8*x^13 - 28*x^12 - 112*x^11 - 386*x^10 - 1084*x^9 + 1550*x^8 + 18704*x^7 + 52532*x^6 + 83536*x^5 + 98107*x^4 + 93808*x^3 + 65772*x^2 + 31856*x + 16631)
 
gp: K = bnfinit(x^16 - 8*x^14 - 8*x^13 - 28*x^12 - 112*x^11 - 386*x^10 - 1084*x^9 + 1550*x^8 + 18704*x^7 + 52532*x^6 + 83536*x^5 + 98107*x^4 + 93808*x^3 + 65772*x^2 + 31856*x + 16631, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 8 x^{13} - 28 x^{12} - 112 x^{11} - 386 x^{10} - 1084 x^{9} + 1550 x^{8} + 18704 x^{7} + 52532 x^{6} + 83536 x^{5} + 98107 x^{4} + 93808 x^{3} + 65772 x^{2} + 31856 x + 16631 \)

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:  \(2963025166336000000000000=2^{32}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.84$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{123} a^{12} + \frac{34}{123} a^{10} + \frac{11}{41} a^{9} + \frac{19}{41} a^{8} - \frac{38}{123} a^{7} + \frac{11}{123} a^{6} - \frac{13}{123} a^{5} + \frac{40}{123} a^{4} + \frac{25}{123} a^{3} + \frac{38}{123} a^{2} - \frac{4}{123} a + \frac{22}{123}$, $\frac{1}{123} a^{13} + \frac{34}{123} a^{11} + \frac{11}{41} a^{10} + \frac{19}{41} a^{9} - \frac{38}{123} a^{8} + \frac{11}{123} a^{7} - \frac{13}{123} a^{6} + \frac{40}{123} a^{5} + \frac{25}{123} a^{4} + \frac{38}{123} a^{3} - \frac{4}{123} a^{2} + \frac{22}{123} a$, $\frac{1}{123} a^{14} + \frac{11}{41} a^{11} + \frac{8}{123} a^{10} - \frac{53}{123} a^{9} + \frac{1}{3} a^{8} + \frac{49}{123} a^{7} + \frac{35}{123} a^{6} - \frac{25}{123} a^{5} + \frac{31}{123} a^{4} + \frac{7}{123} a^{3} - \frac{40}{123} a^{2} + \frac{13}{123} a - \frac{10}{123}$, $\frac{1}{912758477173294801933169463} a^{15} - \frac{769201361712040184587552}{912758477173294801933169463} a^{14} + \frac{2287362104860669300155341}{912758477173294801933169463} a^{13} + \frac{903849307325027890696880}{304252825724431600644389821} a^{12} - \frac{32788507119009738574607495}{912758477173294801933169463} a^{11} + \frac{135945302979289344957925460}{912758477173294801933169463} a^{10} + \frac{431123765978277360095793820}{912758477173294801933169463} a^{9} - \frac{386620780970456315411969282}{912758477173294801933169463} a^{8} + \frac{7211592667393855443490394}{912758477173294801933169463} a^{7} - \frac{271128580232532422058239813}{912758477173294801933169463} a^{6} - \frac{17100795646936523952202490}{912758477173294801933169463} a^{5} + \frac{164133683361159345810324791}{912758477173294801933169463} a^{4} - \frac{179352935306854740864439297}{912758477173294801933169463} a^{3} - \frac{109960629268729794030636602}{304252825724431600644389821} a^{2} + \frac{146768851757895352526023877}{304252825724431600644389821} a - \frac{386033870200839186062760980}{912758477173294801933169463}$

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:  \( \frac{1286423910348914768}{10023593823626962167483} a^{15} - \frac{35982004385547982012}{410967346768705448866803} a^{14} - \frac{261460798519772989768}{410967346768705448866803} a^{13} - \frac{699711161081372238935}{410967346768705448866803} a^{12} - \frac{1288020558341941078208}{410967346768705448866803} a^{11} - \frac{1239936230633520901418}{136989115589568482955601} a^{10} - \frac{22461280681777121487572}{410967346768705448866803} a^{9} - \frac{43702124827024469551604}{410967346768705448866803} a^{8} + \frac{29039609862002659231616}{136989115589568482955601} a^{7} + \frac{299943979861817588921054}{136989115589568482955601} a^{6} + \frac{882235347488938583856948}{136989115589568482955601} a^{5} + \frac{1300899965864286072984731}{136989115589568482955601} a^{4} + \frac{1073993277913030054243664}{136989115589568482955601} a^{3} + \frac{1959059258278034183308802}{410967346768705448866803} a^{2} + \frac{1006124036920034992830740}{410967346768705448866803} a - \frac{223392575641758201585922}{410967346768705448866803} \) (order $10$)
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:  \( 438332.744264 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), 4.0.8000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed