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Properties

Label 16.0.21988037298...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{10}\cdot 19^{4}\cdot 59^{6}$
Root discriminant $44.30$
Ramified primes $2, 5, 19, 59$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group 16T1174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![156301, -4659, 147729, -3854, 54426, 5625, 9689, -722, 1585, 141, -349, 94, 18, -6, 8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 8*x^14 - 6*x^13 + 18*x^12 + 94*x^11 - 349*x^10 + 141*x^9 + 1585*x^8 - 722*x^7 + 9689*x^6 + 5625*x^5 + 54426*x^4 - 3854*x^3 + 147729*x^2 - 4659*x + 156301)
 
gp: K = bnfinit(x^16 - 2*x^15 + 8*x^14 - 6*x^13 + 18*x^12 + 94*x^11 - 349*x^10 + 141*x^9 + 1585*x^8 - 722*x^7 + 9689*x^6 + 5625*x^5 + 54426*x^4 - 3854*x^3 + 147729*x^2 - 4659*x + 156301, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 8 x^{14} - 6 x^{13} + 18 x^{12} + 94 x^{11} - 349 x^{10} + 141 x^{9} + 1585 x^{8} - 722 x^{7} + 9689 x^{6} + 5625 x^{5} + 54426 x^{4} - 3854 x^{3} + 147729 x^{2} - 4659 x + 156301 \)

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:  \(219880372985150440000000000=2^{12}\cdot 5^{10}\cdot 19^{4}\cdot 59^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.30$
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:  $2, 5, 19, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{1045} a^{14} - \frac{3}{95} a^{13} + \frac{7}{1045} a^{12} - \frac{16}{209} a^{11} - \frac{281}{1045} a^{10} + \frac{1}{55} a^{9} + \frac{267}{1045} a^{8} + \frac{414}{1045} a^{7} - \frac{417}{1045} a^{6} + \frac{417}{1045} a^{5} - \frac{82}{1045} a^{4} + \frac{204}{1045} a^{3} - \frac{18}{1045} a^{2} - \frac{5}{11} a - \frac{108}{1045}$, $\frac{1}{181086976141035103973220611533945405} a^{15} + \frac{84534276269937556744319592623833}{181086976141035103973220611533945405} a^{14} - \frac{17580518053809836558986432331714211}{181086976141035103973220611533945405} a^{13} + \frac{1363353479042422107556417893453968}{36217395228207020794644122306789081} a^{12} + \frac{168772608063233250943266387176286}{1906178696221422147086532752988899} a^{11} - \frac{85758613529780949277140182791856987}{181086976141035103973220611533945405} a^{10} - \frac{8494264583132126403796709088409428}{25869568020147871996174373076277915} a^{9} - \frac{5689136759773979276157838544348542}{181086976141035103973220611533945405} a^{8} + \frac{65424856784442691336998667161846918}{181086976141035103973220611533945405} a^{7} + \frac{4500893608566155137819578511400928}{9530893481107110735432663764944495} a^{6} + \frac{1286190970007313069964306123160858}{5173913604029574399234874615255583} a^{5} + \frac{2757813518180658342530613074341847}{16462452376457736724838237412176855} a^{4} + \frac{3758834151809100299957211021236301}{25869568020147871996174373076277915} a^{3} - \frac{19341651838809862976831821547981569}{181086976141035103973220611533945405} a^{2} - \frac{44061610375992800107423360625535558}{181086976141035103973220611533945405} a - \frac{3258245574646762560846567617763801}{36217395228207020794644122306789081}$

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:  \( -1 \) (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:  \( 398217.189935 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 70 conjugacy class representatives for t16n1174 are not computed
Character table for t16n1174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.28025.1, 8.0.741418190000.1, 8.0.62832050000.1, 8.8.3707090950000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.3.1$x^{4} + 177$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
59.4.3.1$x^{4} + 177$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$