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Properties

Label 16.2.18920881198...8367.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,23^{3}\cdot 41^{15}$
Root discriminant $58.52$
Ramified primes $23, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-74332, 256600, -390293, 256782, 19172, -138149, 77730, -5374, -13411, 7653, -1511, -385, 230, -32, 14, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 14*x^14 - 32*x^13 + 230*x^12 - 385*x^11 - 1511*x^10 + 7653*x^9 - 13411*x^8 - 5374*x^7 + 77730*x^6 - 138149*x^5 + 19172*x^4 + 256782*x^3 - 390293*x^2 + 256600*x - 74332)
 
gp: K = bnfinit(x^16 - 7*x^15 + 14*x^14 - 32*x^13 + 230*x^12 - 385*x^11 - 1511*x^10 + 7653*x^9 - 13411*x^8 - 5374*x^7 + 77730*x^6 - 138149*x^5 + 19172*x^4 + 256782*x^3 - 390293*x^2 + 256600*x - 74332, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 14 x^{14} - 32 x^{13} + 230 x^{12} - 385 x^{11} - 1511 x^{10} + 7653 x^{9} - 13411 x^{8} - 5374 x^{7} + 77730 x^{6} - 138149 x^{5} + 19172 x^{4} + 256782 x^{3} - 390293 x^{2} + 256600 x - 74332 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18920881198502041761781608367=-\,23^{3}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.52$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{368} a^{14} - \frac{11}{368} a^{13} + \frac{31}{368} a^{12} - \frac{43}{368} a^{11} - \frac{67}{368} a^{10} - \frac{15}{92} a^{9} - \frac{7}{184} a^{8} + \frac{129}{368} a^{7} - \frac{117}{368} a^{6} - \frac{109}{368} a^{5} + \frac{181}{368} a^{4} + \frac{1}{8} a^{3} - \frac{67}{368} a^{2} - \frac{17}{46} a - \frac{17}{92}$, $\frac{1}{7381084381104764493798017132128} a^{15} + \frac{654657062757098153308201383}{922635547638095561724752141516} a^{14} - \frac{225388730161756427715593137245}{3690542190552382246899008566064} a^{13} + \frac{142946255260139491054819058141}{3690542190552382246899008566064} a^{12} - \frac{125904332161597224106859163841}{1845271095276191123449504283032} a^{11} - \frac{739037692191499737794620843997}{7381084381104764493798017132128} a^{10} - \frac{478127447623464540589771425469}{3690542190552382246899008566064} a^{9} - \frac{160881315243283759272875553489}{7381084381104764493798017132128} a^{8} + \frac{1746202062186802433945018718239}{3690542190552382246899008566064} a^{7} + \frac{619251657193080868153466849385}{1845271095276191123449504283032} a^{6} + \frac{666500701308345348974476583623}{3690542190552382246899008566064} a^{5} + \frac{2137074040223305325781076447149}{7381084381104764493798017132128} a^{4} + \frac{718102590689755431862028907527}{7381084381104764493798017132128} a^{3} - \frac{2188099457818281580622194891049}{7381084381104764493798017132128} a^{2} + \frac{698125615172219996341445851565}{1845271095276191123449504283032} a + \frac{664873687109303825022671408049}{1845271095276191123449504283032}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $8$
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:  \( 155598211.264 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.4479348299263.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
41Data not computed