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Properties

Label 16.0.34552224922...5625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 29^{12}$
Root discriminant $34.17$
Ramified primes $5, 29$
Class number $36$ (GRH)
Class group $[3, 12]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1856, -1624, -2001, 5684, -1956, -5996, 5093, 2722, -2247, -416, 657, 62, 9, 8, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 + 8*x^13 + 9*x^12 + 62*x^11 + 657*x^10 - 416*x^9 - 2247*x^8 + 2722*x^7 + 5093*x^6 - 5996*x^5 - 1956*x^4 + 5684*x^3 - 2001*x^2 - 1624*x + 1856)
 
gp: K = bnfinit(x^16 - 2*x^15 + 6*x^14 + 8*x^13 + 9*x^12 + 62*x^11 + 657*x^10 - 416*x^9 - 2247*x^8 + 2722*x^7 + 5093*x^6 - 5996*x^5 - 1956*x^4 + 5684*x^3 - 2001*x^2 - 1624*x + 1856, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 6 x^{14} + 8 x^{13} + 9 x^{12} + 62 x^{11} + 657 x^{10} - 416 x^{9} - 2247 x^{8} + 2722 x^{7} + 5093 x^{6} - 5996 x^{5} - 1956 x^{4} + 5684 x^{3} - 2001 x^{2} - 1624 x + 1856 \)

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:  \(3455222492240908603515625=5^{10}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.17$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{40} a^{10} + \frac{1}{40} a^{9} - \frac{1}{40} a^{8} - \frac{1}{20} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{17}{40} a^{2} - \frac{1}{4} a + \frac{2}{5}$, $\frac{1}{40} a^{11} - \frac{1}{20} a^{9} - \frac{1}{40} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{7}{40} a^{5} - \frac{9}{20} a^{3} - \frac{7}{40} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{480} a^{12} - \frac{1}{80} a^{11} - \frac{1}{480} a^{10} + \frac{1}{40} a^{9} + \frac{7}{160} a^{8} + \frac{13}{240} a^{7} + \frac{17}{480} a^{6} + \frac{3}{40} a^{5} + \frac{67}{480} a^{4} + \frac{113}{240} a^{3} + \frac{41}{96} a^{2} + \frac{17}{120} a + \frac{1}{15}$, $\frac{1}{480} a^{13} - \frac{1}{480} a^{11} - \frac{1}{80} a^{10} + \frac{3}{160} a^{9} + \frac{1}{60} a^{8} + \frac{53}{480} a^{7} - \frac{1}{80} a^{6} - \frac{41}{480} a^{5} - \frac{1}{15} a^{4} - \frac{227}{480} a^{3} + \frac{1}{240} a^{2} + \frac{11}{30} a - \frac{1}{5}$, $\frac{1}{960} a^{14} - \frac{1}{960} a^{13} - \frac{11}{960} a^{11} - \frac{1}{96} a^{10} - \frac{13}{960} a^{9} + \frac{1}{32} a^{8} - \frac{7}{64} a^{7} + \frac{11}{160} a^{6} - \frac{29}{320} a^{5} + \frac{43}{240} a^{4} + \frac{79}{192} a^{3} + \frac{451}{960} a^{2} + \frac{19}{40} a - \frac{4}{15}$, $\frac{1}{1200652676010782400} a^{15} - \frac{51873479265253}{240130535202156480} a^{14} + \frac{91492530590897}{150081584501347800} a^{13} + \frac{93138134016617}{240130535202156480} a^{12} - \frac{3672011844663913}{600326338005391200} a^{11} + \frac{809359843969813}{80043511734052160} a^{10} - \frac{20376278501402329}{600326338005391200} a^{9} - \frac{20603292637688539}{400217558670260800} a^{8} - \frac{16215830257672541}{200108779335130400} a^{7} - \frac{578054057746319}{11434787390578880} a^{6} - \frac{43636295512640333}{300163169002695600} a^{5} + \frac{7877716842814915}{48026107040431296} a^{4} + \frac{468966233846386379}{1200652676010782400} a^{3} + \frac{32874706020397277}{75040792250673900} a^{2} + \frac{1305374416842109}{37520396125336950} a + \frac{6113706032408458}{18760198062668475}$

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

Class group and class number

$C_{3}\times C_{12}$, which has order $36$ (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:  \( 119871.996246 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.0.121945.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.609725.1 x2, 8.0.371764575625.2 x2, 8.0.371764575625.5, 8.4.64097340625.1 x2

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 8 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$29$29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$