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Properties

Label 16.0.21595140576...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 29^{12}$
Root discriminant $51.10$
Ramified primes $5, 29$
Class number $96$ (GRH)
Class group $[4, 24]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9216, 10368, 3440, -30880, 27740, -28376, 38607, -39130, 30350, -18460, 8917, -3474, 1125, -280, 60, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1125*x^12 - 3474*x^11 + 8917*x^10 - 18460*x^9 + 30350*x^8 - 39130*x^7 + 38607*x^6 - 28376*x^5 + 27740*x^4 - 30880*x^3 + 3440*x^2 + 10368*x + 9216)
 
gp: K = bnfinit(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1125*x^12 - 3474*x^11 + 8917*x^10 - 18460*x^9 + 30350*x^8 - 39130*x^7 + 38607*x^6 - 28376*x^5 + 27740*x^4 - 30880*x^3 + 3440*x^2 + 10368*x + 9216, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1125 x^{12} - 3474 x^{11} + 8917 x^{10} - 18460 x^{9} + 30350 x^{8} - 39130 x^{7} + 38607 x^{6} - 28376 x^{5} + 27740 x^{4} - 30880 x^{3} + 3440 x^{2} + 10368 x + 9216 \)

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:  \(2159514057650567877197265625=5^{14}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{20} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5} + \frac{1}{10} a^{3} - \frac{3}{20} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{20} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{10} a^{5} + \frac{1}{10} a^{4} - \frac{1}{4} a^{3} + \frac{1}{10} a^{2} + \frac{1}{10} a + \frac{1}{5}$, $\frac{1}{80} a^{10} - \frac{1}{80} a^{9} + \frac{7}{40} a^{7} - \frac{1}{5} a^{6} + \frac{1}{40} a^{5} + \frac{13}{80} a^{4} - \frac{3}{16} a^{3} - \frac{1}{40} a^{2} - \frac{3}{20} a$, $\frac{1}{400} a^{11} + \frac{1}{200} a^{10} + \frac{1}{80} a^{9} - \frac{1}{40} a^{8} - \frac{3}{40} a^{7} + \frac{37}{200} a^{6} + \frac{83}{400} a^{5} - \frac{1}{10} a^{4} - \frac{7}{16} a^{3} - \frac{9}{40} a^{2} + \frac{21}{100} a - \frac{2}{25}$, $\frac{1}{2400} a^{12} - \frac{3}{800} a^{10} - \frac{1}{48} a^{9} - \frac{1}{240} a^{8} - \frac{223}{1200} a^{7} - \frac{29}{480} a^{6} + \frac{47}{1200} a^{5} - \frac{7}{160} a^{4} + \frac{7}{80} a^{3} - \frac{49}{600} a^{2} + \frac{13}{30} a - \frac{1}{25}$, $\frac{1}{2400} a^{13} - \frac{1}{800} a^{11} - \frac{1}{300} a^{10} - \frac{1}{240} a^{9} - \frac{13}{1200} a^{8} + \frac{23}{96} a^{7} - \frac{91}{1200} a^{6} - \frac{169}{800} a^{5} + \frac{3}{20} a^{4} + \frac{29}{150} a^{3} - \frac{5}{12} a^{2} - \frac{9}{50} a + \frac{3}{25}$, $\frac{1}{48000} a^{14} - \frac{7}{48000} a^{13} + \frac{3}{16000} a^{12} + \frac{37}{48000} a^{11} + \frac{83}{24000} a^{10} + \frac{133}{6000} a^{9} + \frac{877}{48000} a^{8} - \frac{4399}{48000} a^{7} + \frac{5363}{48000} a^{6} + \frac{2463}{16000} a^{5} + \frac{157}{1500} a^{4} + \frac{1361}{4000} a^{3} - \frac{281}{750} a^{2} - \frac{489}{1000} a - \frac{29}{125}$, $\frac{1}{55248000} a^{15} + \frac{71}{6906000} a^{14} + \frac{43}{6906000} a^{13} - \frac{9}{4604000} a^{12} + \frac{11867}{18416000} a^{11} + \frac{60397}{27624000} a^{10} + \frac{411759}{18416000} a^{9} + \frac{36743}{4604000} a^{8} - \frac{593971}{27624000} a^{7} - \frac{1965341}{9208000} a^{6} + \frac{2988019}{55248000} a^{5} + \frac{2257933}{13812000} a^{4} + \frac{1898913}{4604000} a^{3} - \frac{114311}{1726500} a^{2} - \frac{1247831}{3453000} a - \frac{10829}{28775}$

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

Class group and class number

$C_{4}\times C_{24}$, which has order $96$ (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:  \( 5930917.41967 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{145}) \), 4.4.3048625.2, 4.0.105125.1, 4.0.609725.1, 8.0.1602433515625.2, 8.0.1602433515625.1, 8.0.9294114390625.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$