↑

Properties

Label 16.8.72976987184...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}$
Root discriminant $98.05$
Ramified primes $2, 5, 61, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T864

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3884821, 36216282, 60816943, 24072280, -8196550, 864008, 1869614, -1577310, 386624, -34470, -53739, 26866, -4430, 440, -33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 33*x^14 + 440*x^13 - 4430*x^12 + 26866*x^11 - 53739*x^10 - 34470*x^9 + 386624*x^8 - 1577310*x^7 + 1869614*x^6 + 864008*x^5 - 8196550*x^4 + 24072280*x^3 + 60816943*x^2 + 36216282*x + 3884821)
 
gp: K = bnfinit(x^16 - 6*x^15 - 33*x^14 + 440*x^13 - 4430*x^12 + 26866*x^11 - 53739*x^10 - 34470*x^9 + 386624*x^8 - 1577310*x^7 + 1869614*x^6 + 864008*x^5 - 8196550*x^4 + 24072280*x^3 + 60816943*x^2 + 36216282*x + 3884821, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 33 x^{14} + 440 x^{13} - 4430 x^{12} + 26866 x^{11} - 53739 x^{10} - 34470 x^{9} + 386624 x^{8} - 1577310 x^{7} + 1869614 x^{6} + 864008 x^{5} - 8196550 x^{4} + 24072280 x^{3} + 60816943 x^{2} + 36216282 x + 3884821 \)

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:  \(72976987184482631056000000000000=2^{16}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.05$
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:  $2, 5, 61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3832229460275546852854573622281021435597566745874524516770057} a^{15} + \frac{139385607273722018496523628853862326734763645513071422833909}{1277409820091848950951524540760340478532522248624841505590019} a^{14} + \frac{151745549136224119671571976189627135491501419765199708042083}{1277409820091848950951524540760340478532522248624841505590019} a^{13} - \frac{243551046809689392379135348011306553421220326436457803658375}{3832229460275546852854573622281021435597566745874524516770057} a^{12} + \frac{392156890609560052642510926124898560527858896081103808089724}{1277409820091848950951524540760340478532522248624841505590019} a^{11} + \frac{582673441618774780797028386284258864580857861268256597574533}{1277409820091848950951524540760340478532522248624841505590019} a^{10} + \frac{503939435324440305641431825184578145943108330495268623632002}{1277409820091848950951524540760340478532522248624841505590019} a^{9} + \frac{3553169385096919480901997058601822587183900039755409257070}{19257434473746466597259163931060409224108375607409670938543} a^{8} + \frac{277190856855569010356655926872035321225897754746401718223841}{3832229460275546852854573622281021435597566745874524516770057} a^{7} + \frac{1014365535975577236634943915040320199593521323760186466779688}{3832229460275546852854573622281021435597566745874524516770057} a^{6} + \frac{1551623437922125325490726103976385603911630026409375579085173}{3832229460275546852854573622281021435597566745874524516770057} a^{5} - \frac{1181189117551914914276846427232542023162586221254312206871081}{3832229460275546852854573622281021435597566745874524516770057} a^{4} - \frac{1347466942940664200696191037390314141121264456273856746130541}{3832229460275546852854573622281021435597566745874524516770057} a^{3} - \frac{36450519049506787207464051448806713411340616570405276567318}{3832229460275546852854573622281021435597566745874524516770057} a^{2} - \frac{27633346472053408190801324130365844260196450817192234049626}{1277409820091848950951524540760340478532522248624841505590019} a + \frac{1026610360623701680036153099196610304965229326848440518175709}{3832229460275546852854573622281021435597566745874524516770057}$

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:  \( 9001254201.79 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T864:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n864
Character table for t16n864 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
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}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$