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Properties

Label 16.8.51792035927...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{10}\cdot 29^{6}$
Root discriminant $62.32$
Ramified primes $2, 3, 5, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T293)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7381, 360514, 1288616, 1519488, 694901, -115786, -291002, -47226, 47209, 546, -4460, 100, -505, 54, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 + 54*x^13 - 505*x^12 + 100*x^11 - 4460*x^10 + 546*x^9 + 47209*x^8 - 47226*x^7 - 291002*x^6 - 115786*x^5 + 694901*x^4 + 1519488*x^3 + 1288616*x^2 + 360514*x + 7381)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 + 54*x^13 - 505*x^12 + 100*x^11 - 4460*x^10 + 546*x^9 + 47209*x^8 - 47226*x^7 - 291002*x^6 - 115786*x^5 + 694901*x^4 + 1519488*x^3 + 1288616*x^2 + 360514*x + 7381, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} + 54 x^{13} - 505 x^{12} + 100 x^{11} - 4460 x^{10} + 546 x^{9} + 47209 x^{8} - 47226 x^{7} - 291002 x^{6} - 115786 x^{5} + 694901 x^{4} + 1519488 x^{3} + 1288616 x^{2} + 360514 x + 7381 \)

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:  \(51792035927746314240000000000=2^{24}\cdot 3^{12}\cdot 5^{10}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.32$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} + \frac{1}{3} a^{7} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{63} a^{13} - \frac{1}{63} a^{12} - \frac{2}{21} a^{11} + \frac{5}{63} a^{10} + \frac{10}{63} a^{9} - \frac{1}{21} a^{8} + \frac{2}{9} a^{7} + \frac{19}{63} a^{6} + \frac{3}{7} a^{5} + \frac{13}{63} a^{4} - \frac{4}{9} a^{3} - \frac{2}{7} a^{2} - \frac{23}{63} a - \frac{10}{63}$, $\frac{1}{441} a^{14} - \frac{2}{441} a^{13} + \frac{1}{49} a^{12} - \frac{73}{441} a^{11} + \frac{26}{441} a^{10} - \frac{23}{147} a^{9} - \frac{67}{441} a^{8} - \frac{58}{441} a^{7} + \frac{40}{147} a^{6} + \frac{16}{63} a^{5} + \frac{211}{441} a^{4} + \frac{71}{147} a^{3} - \frac{47}{441} a^{2} + \frac{34}{441} a - \frac{55}{147}$, $\frac{1}{566435157832952220109899620261978294404383} a^{15} - \frac{498203995960879042847593082917038156592}{566435157832952220109899620261978294404383} a^{14} + \frac{916815923847888864011228657925289569281}{566435157832952220109899620261978294404383} a^{13} + \frac{7404647861959883085637574808209042776234}{188811719277650740036633206753992764801461} a^{12} - \frac{11588129874125141578415942880295459153646}{566435157832952220109899620261978294404383} a^{11} - \frac{21658398694887694507271055902040418807496}{566435157832952220109899620261978294404383} a^{10} + \frac{72469627100831632467379320227830965640240}{566435157832952220109899620261978294404383} a^{9} - \frac{81902758375700661086003956552631012967083}{566435157832952220109899620261978294404383} a^{8} + \frac{124554494705196639625194529781081829072550}{566435157832952220109899620261978294404383} a^{7} + \frac{8190919328573606844997689586472119942154}{62937239759216913345544402251330921600487} a^{6} + \frac{129451407732225570339313494960818623577300}{566435157832952220109899620261978294404383} a^{5} + \frac{15371035485371213733376524532504426886011}{51494105257541110919081783660179844945853} a^{4} - \frac{165856719362261297939130702211858625313301}{566435157832952220109899620261978294404383} a^{3} - \frac{149161250047893156877268933848257599057311}{566435157832952220109899620261978294404383} a^{2} + \frac{96726415551894175182111860969649498189518}{566435157832952220109899620261978294404383} a + \frac{5188205085112720884242572540834607900690}{51494105257541110919081783660179844945853}$

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:  $11$
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:  \( 153378952.713 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T293):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), 4.4.104400.1, \(\Q(\sqrt{3}, \sqrt{5})\), 4.4.725.1, 8.8.10899360000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$