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Properties

Label 16.0.25576314038...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 29^{6}$
Root discriminant $38.72$
Ramified primes $2, 3, 5, 29$
Class number $30$ (GRH)
Class group $[30]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8875, 32200, 34230, 9020, 1049, 7572, 12568, 2536, 398, -844, 474, -272, 157, -48, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 48*x^13 + 157*x^12 - 272*x^11 + 474*x^10 - 844*x^9 + 398*x^8 + 2536*x^7 + 12568*x^6 + 7572*x^5 + 1049*x^4 + 9020*x^3 + 34230*x^2 + 32200*x + 8875)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 48*x^13 + 157*x^12 - 272*x^11 + 474*x^10 - 844*x^9 + 398*x^8 + 2536*x^7 + 12568*x^6 + 7572*x^5 + 1049*x^4 + 9020*x^3 + 34230*x^2 + 32200*x + 8875, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 48 x^{13} + 157 x^{12} - 272 x^{11} + 474 x^{10} - 844 x^{9} + 398 x^{8} + 2536 x^{7} + 12568 x^{6} + 7572 x^{5} + 1049 x^{4} + 9020 x^{3} + 34230 x^{2} + 32200 x + 8875 \)

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:  \(25576314038393241600000000=2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.72$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{10} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{6} - \frac{1}{2} a^{4} - \frac{2}{5} a^{2}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{7} - \frac{1}{2} a^{5} - \frac{2}{5} a^{3}$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{8} - \frac{1}{5} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{20} a^{13} + \frac{1}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{4} a$, $\frac{1}{200} a^{14} + \frac{1}{50} a^{13} - \frac{1}{40} a^{12} - \frac{1}{25} a^{11} - \frac{1}{100} a^{10} - \frac{1}{25} a^{9} - \frac{1}{50} a^{7} + \frac{3}{100} a^{6} - \frac{19}{50} a^{5} - \frac{1}{20} a^{4} + \frac{9}{25} a^{3} + \frac{11}{40} a^{2} - \frac{3}{10} a + \frac{3}{8}$, $\frac{1}{313199123324421213824666200} a^{15} + \frac{144558508326456547424169}{313199123324421213824666200} a^{14} - \frac{1243818342790170625535217}{62639824664884242764933240} a^{13} + \frac{5421826675290610242922117}{313199123324421213824666200} a^{12} - \frac{4820647638882472886056631}{156599561662210606912333100} a^{11} + \frac{5524276324549949724660621}{156599561662210606912333100} a^{10} + \frac{76553725250862845882681}{15659956166221060691233310} a^{9} + \frac{11916559432694344610154429}{78299780831105303456166550} a^{8} + \frac{619119080321538875861361}{2954708710607747300232700} a^{7} + \frac{36514107813836181488671317}{156599561662210606912333100} a^{6} - \frac{690075686056086391219269}{6263982466488424276493324} a^{5} + \frac{17071941815811878846927501}{156599561662210606912333100} a^{4} - \frac{3050559852115134622432969}{62639824664884242764933240} a^{3} + \frac{28165577023508351377792963}{62639824664884242764933240} a^{2} - \frac{5804222350426137762049637}{12527964932976848552986648} a - \frac{4575974468468280248093127}{12527964932976848552986648}$

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

Class group and class number

$C_{30}$, which has order $30$ (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:  \( 187751.023506 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{30}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6}) \), 4.0.46400.1, 4.0.6525.1, 4.0.83520.3, \(\Q(\sqrt{5}, \sqrt{6})\), 4.0.83520.4, 8.0.174389760000.18, 8.0.174389760000.15, 8.8.174389760000.1

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 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$