↑

Properties

Label 16.0.32349497931...001.18
Degree $16$
Signature $[0, 8]$
Discriminant $31^{8}\cdot 41^{14}$
Root discriminant $143.50$
Ramified primes $31, 41$
Class number $480$ (GRH)
Class group $[2, 2, 120]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![484313617, -977305750, 1071727507, -759633956, 363624245, -110186296, 14973336, 2807348, -1631495, 192390, 49248, -17034, 674, 432, -54, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 54*x^14 + 432*x^13 + 674*x^12 - 17034*x^11 + 49248*x^10 + 192390*x^9 - 1631495*x^8 + 2807348*x^7 + 14973336*x^6 - 110186296*x^5 + 363624245*x^4 - 759633956*x^3 + 1071727507*x^2 - 977305750*x + 484313617)
 
gp: K = bnfinit(x^16 - 4*x^15 - 54*x^14 + 432*x^13 + 674*x^12 - 17034*x^11 + 49248*x^10 + 192390*x^9 - 1631495*x^8 + 2807348*x^7 + 14973336*x^6 - 110186296*x^5 + 363624245*x^4 - 759633956*x^3 + 1071727507*x^2 - 977305750*x + 484313617, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 54 x^{14} + 432 x^{13} + 674 x^{12} - 17034 x^{11} + 49248 x^{10} + 192390 x^{9} - 1631495 x^{8} + 2807348 x^{7} + 14973336 x^{6} - 110186296 x^{5} + 363624245 x^{4} - 759633956 x^{3} + 1071727507 x^{2} - 977305750 x + 484313617 \)

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:  \(32349497931606921267167334562710001=31^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.50$
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 Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{82} a^{8} - \frac{1}{41} a^{7} + \frac{6}{41} a^{6} - \frac{3}{41} a^{5} + \frac{7}{82} a^{4} - \frac{13}{41} a^{3} - \frac{7}{82} a^{2} + \frac{21}{82} a - \frac{25}{82}$, $\frac{1}{82} a^{9} + \frac{4}{41} a^{7} + \frac{9}{41} a^{6} - \frac{5}{82} a^{5} - \frac{6}{41} a^{4} + \frac{23}{82} a^{3} + \frac{7}{82} a^{2} + \frac{17}{82} a + \frac{16}{41}$, $\frac{1}{82} a^{10} + \frac{17}{41} a^{7} - \frac{19}{82} a^{6} + \frac{18}{41} a^{5} - \frac{33}{82} a^{4} - \frac{31}{82} a^{3} - \frac{9}{82} a^{2} + \frac{14}{41} a + \frac{18}{41}$, $\frac{1}{82} a^{11} - \frac{33}{82} a^{7} + \frac{19}{41} a^{6} + \frac{7}{82} a^{5} - \frac{23}{82} a^{4} - \frac{27}{82} a^{3} + \frac{10}{41} a^{2} - \frac{11}{41} a + \frac{15}{41}$, $\frac{1}{3034} a^{12} - \frac{7}{1517} a^{11} - \frac{3}{1517} a^{10} - \frac{2}{1517} a^{9} + \frac{15}{3034} a^{8} + \frac{84}{1517} a^{7} - \frac{399}{3034} a^{6} - \frac{113}{3034} a^{5} - \frac{1173}{3034} a^{4} - \frac{337}{1517} a^{3} - \frac{716}{1517} a^{2} + \frac{596}{1517} a - \frac{531}{1517}$, $\frac{1}{3034} a^{13} - \frac{17}{3034} a^{11} - \frac{7}{1517} a^{10} - \frac{2}{1517} a^{9} + \frac{4}{1517} a^{8} - \frac{300}{1517} a^{7} - \frac{815}{3034} a^{6} + \frac{5}{74} a^{5} + \frac{479}{3034} a^{4} + \frac{708}{1517} a^{3} - \frac{837}{3034} a^{2} + \frac{419}{3034} a + \frac{373}{1517}$, $\frac{1}{3034} a^{14} + \frac{7}{3034} a^{11} + \frac{5}{3034} a^{10} + \frac{7}{1517} a^{9} - \frac{6}{1517} a^{8} + \frac{114}{1517} a^{7} + \frac{415}{3034} a^{6} - \frac{1035}{3034} a^{5} + \frac{302}{1517} a^{4} + \frac{655}{3034} a^{3} - \frac{319}{3034} a^{2} - \frac{635}{3034} a - \frac{109}{3034}$, $\frac{1}{10793179400477074717923222254919256474} a^{15} + \frac{488345555979178920230005295365029}{10793179400477074717923222254919256474} a^{14} - \frac{688338365428873964537346876020066}{5396589700238537358961611127459628237} a^{13} + \frac{902065451089584532611976298263565}{10793179400477074717923222254919256474} a^{12} + \frac{20918092627281540239016596283466974}{5396589700238537358961611127459628237} a^{11} + \frac{50878869371161331319099139931011105}{10793179400477074717923222254919256474} a^{10} + \frac{31705086892504923427440096481352496}{5396589700238537358961611127459628237} a^{9} - \frac{59640234099882583962302143840256539}{10793179400477074717923222254919256474} a^{8} - \frac{1294658079259946306142658673453662297}{10793179400477074717923222254919256474} a^{7} - \frac{1557727583015097527307884928454435635}{5396589700238537358961611127459628237} a^{6} - \frac{4172872170345403252429735529548887579}{10793179400477074717923222254919256474} a^{5} + \frac{1840995889066255894657768931158056597}{5396589700238537358961611127459628237} a^{4} + \frac{1885446542806123724698474799524434478}{5396589700238537358961611127459628237} a^{3} + \frac{504462122400622633421133157415663941}{10793179400477074717923222254919256474} a^{2} + \frac{701844290316694708340455164470231201}{10793179400477074717923222254919256474} a + \frac{1436047850681511516187789231828271099}{5396589700238537358961611127459628237}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{120}$, which has order $480$ (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:  \( 54888838.1481 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-1271}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{-31}, \sqrt{41})\), 4.4.68921.1, 4.0.66233081.2, 8.0.4386821018752561.2, 8.4.187158857199641.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ 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
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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}$