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Properties

Label 16.0.63893255414...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}$
Root discriminant $199.68$
Ramified primes $5, 29, 89, 97$
Class number $26537856$ (GRH)
Class group $[2, 2, 4, 1658616]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7573496420461, -2034948668117, 3208121031533, -723012409264, 492328618856, -85973445953, 31928083438, -3875600877, 971305477, -77823720, 15260697, -747076, 130172, -3253, 571, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 571*x^14 - 3253*x^13 + 130172*x^12 - 747076*x^11 + 15260697*x^10 - 77823720*x^9 + 971305477*x^8 - 3875600877*x^7 + 31928083438*x^6 - 85973445953*x^5 + 492328618856*x^4 - 723012409264*x^3 + 3208121031533*x^2 - 2034948668117*x + 7573496420461)
 
gp: K = bnfinit(x^16 - 5*x^15 + 571*x^14 - 3253*x^13 + 130172*x^12 - 747076*x^11 + 15260697*x^10 - 77823720*x^9 + 971305477*x^8 - 3875600877*x^7 + 31928083438*x^6 - 85973445953*x^5 + 492328618856*x^4 - 723012409264*x^3 + 3208121031533*x^2 - 2034948668117*x + 7573496420461, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 571 x^{14} - 3253 x^{13} + 130172 x^{12} - 747076 x^{11} + 15260697 x^{10} - 77823720 x^{9} + 971305477 x^{8} - 3875600877 x^{7} + 31928083438 x^{6} - 85973445953 x^{5} + 492328618856 x^{4} - 723012409264 x^{3} + 3208121031533 x^{2} - 2034948668117 x + 7573496420461 \)

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:  \(6389325541442866097681925088134765625=5^{12}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $199.68$
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:  $5, 29, 89, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{145} a^{14} - \frac{9}{145} a^{13} - \frac{6}{145} a^{12} - \frac{32}{145} a^{11} + \frac{27}{145} a^{10} - \frac{9}{29} a^{9} + \frac{14}{29} a^{8} - \frac{1}{5} a^{7} - \frac{68}{145} a^{6} - \frac{34}{145} a^{5} + \frac{72}{145} a^{4} + \frac{34}{145} a^{3} - \frac{9}{29} a^{2} - \frac{26}{145} a - \frac{11}{29}$, $\frac{1}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{15} + \frac{2587371127736052138653491817506940478859094719989179438162237120733119245206452496757}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{14} - \frac{449142046828995148986183451210842266201601771490963689160916693410171548519620317204162}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{13} - \frac{9280105814000372046122164435390230122552398892300871439258521281841804625064454174437}{1075173124084060303509723339894586154300442888146545701548965695800136103454784548159881} a^{12} + \frac{276865476228798878467870004327115627952586325645039205846968325252044239459450327434872}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{11} - \frac{760994125347634295064950535527993197126052425295981321426354598654359872133353587347729}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{10} - \frac{1996559392748201111471118454934038295089957151947063255560360924124419591306362663163049}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{9} + \frac{1766863774285468577241262373702705497427205330638395903053144130666637113243972847322981}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{8} - \frac{142924291988240363516100995482360438357525640618366749825931366828922515382608947002777}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{7} + \frac{1797431812782271529777565137184661349879472052586859933663275335098030469568514016725506}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{6} - \frac{513645062576291010241653350143716699741335163034041238831855967708399007989264814464376}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{5} + \frac{645857107832173134967294041016942392895161435857351046675359104195088740910314973073169}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a^{4} - \frac{36601644772948591182062468994963114301214336799843475943703877677095777294744226568786}{1075173124084060303509723339894586154300442888146545701548965695800136103454784548159881} a^{3} + \frac{36549971501261187497663968992739755928231236872886841062670468816730393315186015922554}{185374676566217293708572989636997612810421187611473396818787188931057948871514577268945} a^{2} - \frac{1209193653109594679058511090751417021585017346618707892235341016387181390357875107911196}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405} a + \frac{362575117087940757006217149113660941448674721450856069607257850892291487025019118272102}{5375865620420301517548616699472930771502214440732728507744828479000680517273922740799405}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{1658616}$, which has order $26537856$ (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:  \( 13998.0176198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.1, 4.4.725.1, 4.4.2225.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$