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Properties

Label 16.8.19234841775...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 13^{2}\cdot 29^{6}\cdot 97^{8}$
Root discriminant $160.42$
Ramified primes $5, 13, 29, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-80946349, -1291737162, 696989411, 291344387, -598924089, 108400439, -30238421, -6636719, 6616977, -57699, 23059, 5969, -5329, -3, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 9*x^14 - 3*x^13 - 5329*x^12 + 5969*x^11 + 23059*x^10 - 57699*x^9 + 6616977*x^8 - 6636719*x^7 - 30238421*x^6 + 108400439*x^5 - 598924089*x^4 + 291344387*x^3 + 696989411*x^2 - 1291737162*x - 80946349)
 
gp: K = bnfinit(x^16 - 2*x^15 - 9*x^14 - 3*x^13 - 5329*x^12 + 5969*x^11 + 23059*x^10 - 57699*x^9 + 6616977*x^8 - 6636719*x^7 - 30238421*x^6 + 108400439*x^5 - 598924089*x^4 + 291344387*x^3 + 696989411*x^2 - 1291737162*x - 80946349, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 9 x^{14} - 3 x^{13} - 5329 x^{12} + 5969 x^{11} + 23059 x^{10} - 57699 x^{9} + 6616977 x^{8} - 6636719 x^{7} - 30238421 x^{6} + 108400439 x^{5} - 598924089 x^{4} + 291344387 x^{3} + 696989411 x^{2} - 1291737162 x - 80946349 \)

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:  \(192348417750098090008878000244140625=5^{12}\cdot 13^{2}\cdot 29^{6}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $160.42$
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:  $5, 13, 29, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{145} a^{12} - \frac{7}{145} a^{11} + \frac{13}{145} a^{10} - \frac{12}{145} a^{9} + \frac{7}{145} a^{8} - \frac{54}{145} a^{7} - \frac{34}{145} a^{6} - \frac{10}{29} a^{5} + \frac{53}{145} a^{4} + \frac{1}{29} a^{3} + \frac{2}{5} a^{2} + \frac{71}{145} a - \frac{47}{145}$, $\frac{1}{145} a^{13} - \frac{7}{145} a^{11} - \frac{8}{145} a^{10} + \frac{2}{29} a^{9} - \frac{1}{29} a^{8} - \frac{64}{145} a^{7} - \frac{27}{145} a^{6} + \frac{51}{145} a^{5} - \frac{59}{145} a^{4} + \frac{7}{29} a^{3} - \frac{9}{29} a^{2} + \frac{44}{145} a - \frac{39}{145}$, $\frac{1}{290} a^{14} - \frac{1}{290} a^{13} - \frac{1}{290} a^{12} + \frac{3}{58} a^{11} + \frac{9}{290} a^{10} - \frac{1}{10} a^{9} - \frac{17}{290} a^{8} - \frac{11}{58} a^{7} + \frac{27}{58} a^{6} + \frac{83}{290} a^{5} - \frac{23}{290} a^{4} - \frac{137}{290} a^{3} + \frac{89}{290} a^{2} + \frac{111}{290} a - \frac{127}{290}$, $\frac{1}{721083101750989061550374464730598814659339262278895804198958463610} a^{15} - \frac{1014300410950211089765971358591651660950548160800639620841342821}{721083101750989061550374464730598814659339262278895804198958463610} a^{14} + \frac{836152667346780018029938268023581270199710193050461696687754207}{721083101750989061550374464730598814659339262278895804198958463610} a^{13} - \frac{420992643394938352439467899510915746664628587001817162960491437}{144216620350197812310074892946119762931867852455779160839791692722} a^{12} + \frac{1531568125157328108586884610610745409792565445141325082988951873}{24864934543137553846564636714848234988253078009617096696515809090} a^{11} - \frac{62306063908189038019347603207450366345842088688562803798572173053}{721083101750989061550374464730598814659339262278895804198958463610} a^{10} - \frac{64812184165280809144639577249334539045233436513197570287207225271}{721083101750989061550374464730598814659339262278895804198958463610} a^{9} - \frac{6856881659917818203813634865937549382625849938396667150789037489}{721083101750989061550374464730598814659339262278895804198958463610} a^{8} - \frac{72696838117772375994893134867149385321225479083552138018421551519}{721083101750989061550374464730598814659339262278895804198958463610} a^{7} + \frac{14143744092780563967582981844748860687799374630812022846115888147}{721083101750989061550374464730598814659339262278895804198958463610} a^{6} + \frac{71799007369205155624698137935163470409336856282407689516732603121}{721083101750989061550374464730598814659339262278895804198958463610} a^{5} - \frac{3624968082449108584821256664688929336020658093039848355788194905}{13110601850017982937279535722374523902897077495979923712708335702} a^{4} + \frac{250448381475711483276325745141438487453459203235932245520478893151}{721083101750989061550374464730598814659339262278895804198958463610} a^{3} - \frac{15122930927529142141233764422605111232692582531149750771527228749}{144216620350197812310074892946119762931867852455779160839791692722} a^{2} + \frac{171894982936958763970052385149455105167240374852176208937923466533}{721083101750989061550374464730598814659339262278895804198958463610} a - \frac{2863107057282936997722824628477865506107059733307821837157315200}{6555300925008991468639767861187261951448538747989961856354167851}$

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:  \( 587311848098 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{485}) \), 4.4.725.1, \(\Q(\sqrt{5}, \sqrt{97})\), 4.4.6821525.1, 8.8.46533203325625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$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}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$