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Properties

Label 16.4.46370927160...3125.2
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 11^{6}\cdot 101^{7}$
Root discriminant $61.89$
Ramified primes $5, 11, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1574

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![660911, 398022, 27250, -324980, 198189, -276618, -191977, 121490, -9069, -8933, 2763, -304, 149, -11, 4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 4*x^14 - 11*x^13 + 149*x^12 - 304*x^11 + 2763*x^10 - 8933*x^9 - 9069*x^8 + 121490*x^7 - 191977*x^6 - 276618*x^5 + 198189*x^4 - 324980*x^3 + 27250*x^2 + 398022*x + 660911)
 
gp: K = bnfinit(x^16 - 6*x^15 + 4*x^14 - 11*x^13 + 149*x^12 - 304*x^11 + 2763*x^10 - 8933*x^9 - 9069*x^8 + 121490*x^7 - 191977*x^6 - 276618*x^5 + 198189*x^4 - 324980*x^3 + 27250*x^2 + 398022*x + 660911, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 4 x^{14} - 11 x^{13} + 149 x^{12} - 304 x^{11} + 2763 x^{10} - 8933 x^{9} - 9069 x^{8} + 121490 x^{7} - 191977 x^{6} - 276618 x^{5} + 198189 x^{4} - 324980 x^{3} + 27250 x^{2} + 398022 x + 660911 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(46370927160987469302001953125=5^{12}\cdot 11^{6}\cdot 101^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.89$
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:  $5, 11, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{11} + \frac{2}{11} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} + \frac{4}{11} a^{7} + \frac{5}{11} a^{6} - \frac{2}{11} a^{5} - \frac{2}{11} a^{4} + \frac{4}{11} a^{3} - \frac{1}{11} a^{2} - \frac{5}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{11} + \frac{1}{11} a^{10} + \frac{5}{11} a^{8} - \frac{2}{11} a^{7} + \frac{3}{11} a^{6} - \frac{4}{11} a^{5} + \frac{2}{11} a^{4} + \frac{3}{11} a^{3} + \frac{5}{11} a^{2} - \frac{2}{11} a + \frac{3}{11}$, $\frac{1}{33} a^{14} + \frac{1}{33} a^{13} + \frac{14}{33} a^{11} + \frac{10}{33} a^{10} - \frac{16}{33} a^{9} - \frac{3}{11} a^{8} - \frac{1}{11} a^{7} - \frac{2}{11} a^{6} + \frac{1}{3} a^{5} + \frac{7}{33} a^{4} + \frac{5}{11} a^{3} + \frac{4}{33} a^{2} - \frac{16}{33} a - \frac{1}{3}$, $\frac{1}{250803720247442443399919726301228727950194613177} a^{15} + \frac{83788795663665880344961758742096012451730296}{83601240082480814466639908767076242650064871059} a^{14} + \frac{11215777219666183922288083670709890636708859674}{250803720247442443399919726301228727950194613177} a^{13} + \frac{9744988892821343493645987875677184669541135149}{250803720247442443399919726301228727950194613177} a^{12} - \frac{82866115003968030025007933012085573379854094609}{250803720247442443399919726301228727950194613177} a^{11} + \frac{92715776381467906465265191361000861335201594096}{250803720247442443399919726301228727950194613177} a^{10} - \frac{2877133334357844832806586968281473778591385890}{250803720247442443399919726301228727950194613177} a^{9} + \frac{22653533505751639789348057511809866888271188441}{83601240082480814466639908767076242650064871059} a^{8} + \frac{32452565510383434697323743039460845043691970806}{83601240082480814466639908767076242650064871059} a^{7} + \frac{14914781922692332351996587280539207090966908828}{250803720247442443399919726301228727950194613177} a^{6} - \frac{86365029423370728862544815589495058235758232954}{250803720247442443399919726301228727950194613177} a^{5} + \frac{60494516959489098232734500290565414354603016125}{250803720247442443399919726301228727950194613177} a^{4} + \frac{119439096844715373602850295698375699704217440455}{250803720247442443399919726301228727950194613177} a^{3} + \frac{23464968413410584481550770018763466430349615815}{250803720247442443399919726301228727950194613177} a^{2} - \frac{89429573532644544792761694136572136097040190618}{250803720247442443399919726301228727950194613177} a - \frac{5739652558847538921901547795605545293239951067}{22800338204312949399992702391020793450017692107}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 43974312.8612 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1574:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 88 conjugacy class representatives for t16n1574 are not computed
Character table for t16n1574 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.4.1947912828125.6

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $16$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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}$
$11$11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$101$101.4.3.3$x^{4} + 202$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$