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Properties

Label 16.12.1571275555...5489.2
Degree $16$
Signature $[12, 2]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![139, -3767, -48690, -223555, -235098, 820974, 801331, -617694, -559066, -43795, 33232, 6298, 739, -64, -72, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 72*x^14 - 64*x^13 + 739*x^12 + 6298*x^11 + 33232*x^10 - 43795*x^9 - 559066*x^8 - 617694*x^7 + 801331*x^6 + 820974*x^5 - 235098*x^4 - 223555*x^3 - 48690*x^2 - 3767*x + 139)
 
gp: K = bnfinit(x^16 - 2*x^15 - 72*x^14 - 64*x^13 + 739*x^12 + 6298*x^11 + 33232*x^10 - 43795*x^9 - 559066*x^8 - 617694*x^7 + 801331*x^6 + 820974*x^5 - 235098*x^4 - 223555*x^3 - 48690*x^2 - 3767*x + 139, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 72 x^{14} - 64 x^{13} + 739 x^{12} + 6298 x^{11} + 33232 x^{10} - 43795 x^{9} - 559066 x^{8} - 617694 x^{7} + 801331 x^{6} + 820974 x^{5} - 235098 x^{4} - 223555 x^{3} - 48690 x^{2} - 3767 x + 139 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{46} a^{14} - \frac{11}{46} a^{13} + \frac{1}{46} a^{12} - \frac{17}{46} a^{11} + \frac{15}{46} a^{10} - \frac{19}{46} a^{9} - \frac{15}{46} a^{8} + \frac{5}{46} a^{7} - \frac{5}{46} a^{6} + \frac{1}{46} a^{5} + \frac{17}{46} a^{4} + \frac{17}{46} a^{3} + \frac{11}{46} a^{2} - \frac{7}{46} a - \frac{15}{46}$, $\frac{1}{3566515559942137617866696439446775913433726} a^{15} + \frac{34705349030537853541655213088213840297049}{3566515559942137617866696439446775913433726} a^{14} + \frac{133678757290481118581041860760572633878969}{1783257779971068808933348219723387956716863} a^{13} + \frac{652666341249637411646972349247763989320543}{3566515559942137617866696439446775913433726} a^{12} + \frac{82505387776526903027855779739378389974797}{3566515559942137617866696439446775913433726} a^{11} - \frac{772442073320979919185194583819755766733521}{1783257779971068808933348219723387956716863} a^{10} + \frac{300077024186879425206779516305102041668809}{3566515559942137617866696439446775913433726} a^{9} + \frac{832041705360142294377966687934787519263085}{3566515559942137617866696439446775913433726} a^{8} - \frac{322958094989626456501025337122803447101799}{1783257779971068808933348219723387956716863} a^{7} - \frac{431164855957898327802565211578059398978577}{3566515559942137617866696439446775913433726} a^{6} + \frac{555518259066844202900335768932603790572305}{3566515559942137617866696439446775913433726} a^{5} + \frac{232897212807255202309681983684001460603905}{1783257779971068808933348219723387956716863} a^{4} + \frac{1004723128878748753249929308242917579932717}{3566515559942137617866696439446775913433726} a^{3} + \frac{1106311338848836935876076384017486428369749}{3566515559942137617866696439446775913433726} a^{2} + \frac{95286222178784054850326523363188913373894}{1783257779971068808933348219723387956716863} a - \frac{470639728048857341355379546551862527140043}{1783257779971068808933348219723387956716863}$

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:  $13$
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:  \( 3413144825360 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed