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Properties

Label 16.0.19641717721...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 31^{4}\cdot 859^{4}$
Root discriminant $28.56$
Ramified primes $5, 31, 859$
Class number $2$
Class group $[2]$
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![39019, -20828, 14457, -14510, 14764, -10126, 9714, -7210, 5444, -2775, 890, -154, 57, -20, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 9*x^14 - 20*x^13 + 57*x^12 - 154*x^11 + 890*x^10 - 2775*x^9 + 5444*x^8 - 7210*x^7 + 9714*x^6 - 10126*x^5 + 14764*x^4 - 14510*x^3 + 14457*x^2 - 20828*x + 39019)
 
gp: K = bnfinit(x^16 - 3*x^15 + 9*x^14 - 20*x^13 + 57*x^12 - 154*x^11 + 890*x^10 - 2775*x^9 + 5444*x^8 - 7210*x^7 + 9714*x^6 - 10126*x^5 + 14764*x^4 - 14510*x^3 + 14457*x^2 - 20828*x + 39019, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 9 x^{14} - 20 x^{13} + 57 x^{12} - 154 x^{11} + 890 x^{10} - 2775 x^{9} + 5444 x^{8} - 7210 x^{7} + 9714 x^{6} - 10126 x^{5} + 14764 x^{4} - 14510 x^{3} + 14457 x^{2} - 20828 x + 39019 \)

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:  \(196417177218537844140625=5^{8}\cdot 31^{4}\cdot 859^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.56$
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:  $5, 31, 859$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{1}{5} a^{11} + \frac{1}{3} a^{10} + \frac{2}{15} a^{9} + \frac{1}{5} a^{8} + \frac{1}{3} a^{7} - \frac{7}{15} a^{6} + \frac{2}{15} a^{5} + \frac{1}{3} a^{4} - \frac{1}{5} a^{3} + \frac{7}{15} a^{2} - \frac{7}{15} a + \frac{1}{15}$, $\frac{1}{15} a^{13} - \frac{4}{15} a^{11} + \frac{2}{15} a^{10} - \frac{2}{5} a^{9} - \frac{1}{15} a^{8} - \frac{7}{15} a^{7} - \frac{4}{15} a^{6} - \frac{4}{15} a^{5} - \frac{1}{5} a^{4} - \frac{2}{15} a^{3} - \frac{1}{15} a^{2} - \frac{1}{3} a + \frac{1}{5}$, $\frac{1}{15} a^{14} + \frac{1}{3} a^{11} - \frac{1}{15} a^{10} + \frac{7}{15} a^{9} + \frac{1}{3} a^{8} + \frac{1}{15} a^{7} - \frac{2}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{5} a^{4} + \frac{2}{15} a^{3} - \frac{7}{15} a^{2} + \frac{1}{3} a + \frac{4}{15}$, $\frac{1}{8161661140580917928181934925634045} a^{15} + \frac{152558166245788428499647965691092}{8161661140580917928181934925634045} a^{14} + \frac{97870171164139776048543908114027}{8161661140580917928181934925634045} a^{13} + \frac{250505276218307652489728214830543}{8161661140580917928181934925634045} a^{12} - \frac{867656968675117620053792116001438}{8161661140580917928181934925634045} a^{11} - \frac{1352396349952612159613738654783012}{2720553713526972642727311641878015} a^{10} + \frac{85402080518976966548061709005058}{8161661140580917928181934925634045} a^{9} + \frac{417777271974872453856421021284236}{2720553713526972642727311641878015} a^{8} + \frac{927148267689616075420485262626796}{8161661140580917928181934925634045} a^{7} - \frac{1690910634588940224639808462405618}{8161661140580917928181934925634045} a^{6} + \frac{1324673232124739915909018402420846}{8161661140580917928181934925634045} a^{5} - \frac{2903991500024221164351807057315208}{8161661140580917928181934925634045} a^{4} + \frac{2862306829618546170597279931829129}{8161661140580917928181934925634045} a^{3} + \frac{244587845371929679734701737026122}{1632332228116183585636386985126809} a^{2} + \frac{3406584331887998458638532762770598}{8161661140580917928181934925634045} a + \frac{2634085118947459701270605640146432}{8161661140580917928181934925634045}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 35623.8314936 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1869 are not computed
Character table for t16n1869 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.16643125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
859Data not computed