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Properties

Label 16.0.80039426073...0000.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 29^{8}$
Root discriminant $36.01$
Ramified primes $2, 5, 29$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4001, 20168, 53082, 81152, 81851, 57154, 28265, 10170, 3011, 960, 220, -8, -6, 8, -5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 5*x^14 + 8*x^13 - 6*x^12 - 8*x^11 + 220*x^10 + 960*x^9 + 3011*x^8 + 10170*x^7 + 28265*x^6 + 57154*x^5 + 81851*x^4 + 81152*x^3 + 53082*x^2 + 20168*x + 4001)
 
gp: K = bnfinit(x^16 - 4*x^15 - 5*x^14 + 8*x^13 - 6*x^12 - 8*x^11 + 220*x^10 + 960*x^9 + 3011*x^8 + 10170*x^7 + 28265*x^6 + 57154*x^5 + 81851*x^4 + 81152*x^3 + 53082*x^2 + 20168*x + 4001, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 5 x^{14} + 8 x^{13} - 6 x^{12} - 8 x^{11} + 220 x^{10} + 960 x^{9} + 3011 x^{8} + 10170 x^{7} + 28265 x^{6} + 57154 x^{5} + 81851 x^{4} + 81152 x^{3} + 53082 x^{2} + 20168 x + 4001 \)

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:  \(8003942607376000000000000=2^{16}\cdot 5^{12}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.01$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{6} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{26030551084149377437260579798} a^{15} + \frac{7635207074589207193892665}{839695196262883143137438058} a^{14} + \frac{529522507868750285938717615}{4338425180691562906210096633} a^{13} + \frac{741237045850146291112831076}{4338425180691562906210096633} a^{12} + \frac{840299140078695072511516486}{13015275542074688718630289899} a^{11} - \frac{5652091068695918994053537903}{13015275542074688718630289899} a^{10} + \frac{1886965231762165256900520947}{13015275542074688718630289899} a^{9} + \frac{1316231404187082924766197717}{4338425180691562906210096633} a^{8} + \frac{2750728313704216960521086221}{26030551084149377437260579798} a^{7} + \frac{1310559964226149668335574821}{8676850361383125812420193266} a^{6} + \frac{85781848859812336094509660}{213365172820896536370988359} a^{5} - \frac{169969950300527359859738812}{419847598131441571568719029} a^{4} + \frac{617241232285427983901229239}{8676850361383125812420193266} a^{3} + \frac{12797004020552122667668795739}{26030551084149377437260579798} a^{2} - \frac{8723244307209286600342488257}{26030551084149377437260579798} a + \frac{9757288990017097056962204275}{26030551084149377437260579798}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( -\frac{6510244737717413}{2115798334495456614} a^{15} + \frac{383772036406255}{22750519725757598} a^{14} - \frac{10450893618732460}{1057899167247728307} a^{13} - \frac{3107988334494665}{352633055749242769} a^{12} + \frac{11345775193567245}{352633055749242769} a^{11} - \frac{25347889213887403}{1057899167247728307} a^{10} - \frac{682431740421381635}{1057899167247728307} a^{9} - \frac{2103313489460432120}{1057899167247728307} a^{8} - \frac{13481895582610437485}{2115798334495456614} a^{7} - \frac{46756405729619994605}{2115798334495456614} a^{6} - \frac{317377602820020367}{5780869766381029} a^{5} - \frac{3299542239362235685}{34125779588636397} a^{4} - \frac{80974424930234909815}{705266111498485538} a^{3} - \frac{193159292338344866965}{2115798334495456614} a^{2} - \frac{88318474726387155715}{2115798334495456614} a - \frac{21190498447983334541}{2115798334495456614} \) (order $4$)
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:  \( 587718.81982 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.4.58000.1, 4.0.3625.1, \(\Q(i, \sqrt{5})\), 8.0.3364000000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$