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Properties

Label 16.0.12967247986...8961.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 43^{4}$
Root discriminant $66.00$
Ramified primes $41, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![636931, -823177, 619715, 197266, -455538, 350155, -58067, -79156, 80287, -36523, 9075, -450, -454, 118, 0, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 118*x^13 - 454*x^12 - 450*x^11 + 9075*x^10 - 36523*x^9 + 80287*x^8 - 79156*x^7 - 58067*x^6 + 350155*x^5 - 455538*x^4 + 197266*x^3 + 619715*x^2 - 823177*x + 636931)
 
gp: K = bnfinit(x^16 - 6*x^15 + 118*x^13 - 454*x^12 - 450*x^11 + 9075*x^10 - 36523*x^9 + 80287*x^8 - 79156*x^7 - 58067*x^6 + 350155*x^5 - 455538*x^4 + 197266*x^3 + 619715*x^2 - 823177*x + 636931, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 118 x^{13} - 454 x^{12} - 450 x^{11} + 9075 x^{10} - 36523 x^{9} + 80287 x^{8} - 79156 x^{7} - 58067 x^{6} + 350155 x^{5} - 455538 x^{4} + 197266 x^{3} + 619715 x^{2} - 823177 x + 636931 \)

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:  \(129672479863204507348386828961=41^{14}\cdot 43^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.00$
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:  $41, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} + \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{612208} a^{14} - \frac{17785}{612208} a^{13} + \frac{2909}{306104} a^{12} - \frac{72051}{612208} a^{11} - \frac{37927}{612208} a^{10} - \frac{127859}{306104} a^{9} - \frac{35361}{153052} a^{8} - \frac{61225}{612208} a^{7} + \frac{125195}{306104} a^{6} - \frac{141781}{612208} a^{5} - \frac{109403}{306104} a^{4} + \frac{127275}{306104} a^{3} + \frac{9375}{306104} a^{2} - \frac{121259}{306104} a - \frac{256493}{612208}$, $\frac{1}{4077862309853507944817099155683171568} a^{15} + \frac{961910417171791432888095048299}{4077862309853507944817099155683171568} a^{14} + \frac{27824829185733792285070626973532117}{2038931154926753972408549577841585784} a^{13} + \frac{164159243636239987031760520022858027}{4077862309853507944817099155683171568} a^{12} + \frac{2811893815748873525304538183879913}{4077862309853507944817099155683171568} a^{11} + \frac{122859344392151226211320277343930375}{509732788731688493102137394460396446} a^{10} + \frac{60677332928961866626114660023452375}{1019465577463376986204274788920792892} a^{9} + \frac{1309659680971811615728052064024705055}{4077862309853507944817099155683171568} a^{8} + \frac{735864948638039775433582808901319735}{2038931154926753972408549577841585784} a^{7} - \frac{685407145608362378609133936427991221}{4077862309853507944817099155683171568} a^{6} - \frac{234975192072045664030317275696881913}{1019465577463376986204274788920792892} a^{5} + \frac{31086694029395890480962779168354135}{1019465577463376986204274788920792892} a^{4} - \frac{64793189565579107344116560237928881}{1019465577463376986204274788920792892} a^{3} - \frac{412050449908880767202721997882431647}{2038931154926753972408549577841585784} a^{2} - \frac{137003266586518745460504977634319387}{4077862309853507944817099155683171568} a + \frac{812733884929440545107046746278436249}{2038931154926753972408549577841585784}$

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:  $7$
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:  \( 53320614.8559 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.2.2963603.1, 4.4.68921.1, 4.2.72283.1, 8.4.360100652405969.1, 8.0.194754273881.1, 8.4.8782942741609.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
41Data not computed
$43$43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$