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Properties

Label 16.0.20392810905...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{4}\cdot 5^{12}\cdot 7^{4}$
Root discriminant $28.63$
Ramified primes $2, 3, 5, 7$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_2^3:(C_2\times C_4)$ (as 16T68)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, -8692, 21608, -31104, 32902, -26792, 18568, -10592, 5580, -2152, 816, -224, 77, -4, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 4*x^13 + 77*x^12 - 224*x^11 + 816*x^10 - 2152*x^9 + 5580*x^8 - 10592*x^7 + 18568*x^6 - 26792*x^5 + 32902*x^4 - 31104*x^3 + 21608*x^2 - 8692*x + 1681)
 
gp: K = bnfinit(x^16 + 8*x^14 - 4*x^13 + 77*x^12 - 224*x^11 + 816*x^10 - 2152*x^9 + 5580*x^8 - 10592*x^7 + 18568*x^6 - 26792*x^5 + 32902*x^4 - 31104*x^3 + 21608*x^2 - 8692*x + 1681, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 4 x^{13} + 77 x^{12} - 224 x^{11} + 816 x^{10} - 2152 x^{9} + 5580 x^{8} - 10592 x^{7} + 18568 x^{6} - 26792 x^{5} + 32902 x^{4} - 31104 x^{3} + 21608 x^{2} - 8692 x + 1681 \)

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:  \(203928109056000000000000=2^{32}\cdot 3^{4}\cdot 5^{12}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.63$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{140015152609448601811431398253} a^{15} - \frac{320597288675640027568101992}{3415003722181673214912960933} a^{14} + \frac{4086458019589446116770190447}{140015152609448601811431398253} a^{13} - \frac{2960709655344995775245805362}{140015152609448601811431398253} a^{12} - \frac{49863498597931381144451001412}{140015152609448601811431398253} a^{11} + \frac{60352005186207799544437992749}{140015152609448601811431398253} a^{10} - \frac{24821849026065474145794239081}{140015152609448601811431398253} a^{9} - \frac{11389231656334183264796641502}{140015152609448601811431398253} a^{8} + \frac{6257837322062322921320525647}{140015152609448601811431398253} a^{7} + \frac{11685295143281911780613612486}{140015152609448601811431398253} a^{6} - \frac{21680996085514680849776290019}{140015152609448601811431398253} a^{5} - \frac{40377713905502662875827325310}{140015152609448601811431398253} a^{4} - \frac{53975431122421805231338821355}{140015152609448601811431398253} a^{3} - \frac{13839038362939799439301777328}{46671717536482867270477132751} a^{2} - \frac{31372467271178105018182260962}{140015152609448601811431398253} a - \frac{1187495445375341889741187843}{3415003722181673214912960933}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( -\frac{38357752525953503147132288}{46671717536482867270477132751} a^{15} - \frac{312331514840991465465160}{1138334574060557738304320311} a^{14} - \frac{921573156607523171637817348}{140015152609448601811431398253} a^{13} + \frac{51206173083362387403265151}{46671717536482867270477132751} a^{12} - \frac{8753337759222014902286791264}{140015152609448601811431398253} a^{11} + \frac{22719808759441368163468810288}{140015152609448601811431398253} a^{10} - \frac{28590428530040730788491034536}{46671717536482867270477132751} a^{9} + \frac{71984627462971259713902655738}{46671717536482867270477132751} a^{8} - \frac{563427103859841920045045001568}{140015152609448601811431398253} a^{7} + \frac{1008540931087220968300728269896}{140015152609448601811431398253} a^{6} - \frac{1750211427449793267295784360792}{140015152609448601811431398253} a^{5} + \frac{2404746841219914943183165400962}{140015152609448601811431398253} a^{4} - \frac{2850050003492886120258524252416}{140015152609448601811431398253} a^{3} + \frac{2360452938795113637166981663016}{140015152609448601811431398253} a^{2} - \frac{1490124694861208251018926256100}{140015152609448601811431398253} a + \frac{8504553043592800439317958416}{3415003722181673214912960933} \) (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:  \( 36969.9190932 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:(C_2\times C_4)$ (as 16T68):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 34 conjugacy class representatives for $C_2^3:(C_2\times C_4)$
Character table for $C_2^3:(C_2\times C_4)$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.18063360000.1, 8.0.451584000000.15, 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 R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$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$$[\ ]_{4}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$