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Properties

Label 16.0.11819246862...7024.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{8}$
Root discriminant $17.97$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -24, 196, -656, 1408, -2160, 2340, -1856, 976, -256, 28, 72, -40, -8, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 8*x^13 - 40*x^12 + 72*x^11 + 28*x^10 - 256*x^9 + 976*x^8 - 1856*x^7 + 2340*x^6 - 2160*x^5 + 1408*x^4 - 656*x^3 + 196*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 8*x^13 - 40*x^12 + 72*x^11 + 28*x^10 - 256*x^9 + 976*x^8 - 1856*x^7 + 2340*x^6 - 2160*x^5 + 1408*x^4 - 656*x^3 + 196*x^2 - 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 8 x^{13} - 40 x^{12} + 72 x^{11} + 28 x^{10} - 256 x^{9} + 976 x^{8} - 1856 x^{7} + 2340 x^{6} - 2160 x^{5} + 1408 x^{4} - 656 x^{3} + 196 x^{2} - 24 x + 1 \)

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:  \(118192468620711297024=2^{54}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.97$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} - \frac{5}{17} a^{9} + \frac{3}{17} a^{8} + \frac{3}{17} a^{6} + \frac{2}{17} a^{5} - \frac{8}{17} a^{4} - \frac{4}{17} a^{2} + \frac{3}{17} a + \frac{5}{17}$, $\frac{1}{17} a^{11} - \frac{5}{17} a^{9} - \frac{2}{17} a^{8} + \frac{3}{17} a^{7} + \frac{2}{17} a^{5} - \frac{6}{17} a^{4} - \frac{4}{17} a^{3} + \frac{3}{17} a + \frac{8}{17}$, $\frac{1}{17} a^{12} + \frac{7}{17} a^{9} + \frac{1}{17} a^{8} + \frac{4}{17} a^{5} + \frac{7}{17} a^{4} + \frac{6}{17} a + \frac{8}{17}$, $\frac{1}{17} a^{13} + \frac{2}{17} a^{9} - \frac{4}{17} a^{8} - \frac{7}{17} a^{5} + \frac{5}{17} a^{4} + \frac{4}{17} a - \frac{1}{17}$, $\frac{1}{17} a^{14} + \frac{6}{17} a^{9} - \frac{6}{17} a^{8} + \frac{4}{17} a^{6} + \frac{1}{17} a^{5} - \frac{1}{17} a^{4} - \frac{5}{17} a^{2} - \frac{7}{17} a + \frac{7}{17}$, $\frac{1}{10011279010049} a^{15} - \frac{3371877141}{141003929719} a^{14} - \frac{207092760433}{10011279010049} a^{13} + \frac{124376895470}{10011279010049} a^{12} - \frac{134610912952}{10011279010049} a^{11} - \frac{58858877701}{10011279010049} a^{10} - \frac{614020599052}{10011279010049} a^{9} + \frac{2829793352393}{10011279010049} a^{8} - \frac{4093723127197}{10011279010049} a^{7} - \frac{2799286717824}{10011279010049} a^{6} + \frac{2782587376367}{10011279010049} a^{5} - \frac{4125599995519}{10011279010049} a^{4} + \frac{1192420874382}{10011279010049} a^{3} + \frac{22148215012}{588898765297} a^{2} + \frac{2097496330420}{10011279010049} a + \frac{2613321612496}{10011279010049}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{396205164960}{588898765297} a^{15} - \frac{44785681030}{8294348807} a^{14} + \frac{7955046201333}{588898765297} a^{13} - \frac{3005709842981}{588898765297} a^{12} - \frac{971296358495}{34641103841} a^{11} + \frac{28776720723641}{588898765297} a^{10} + \frac{12375493681770}{588898765297} a^{9} - \frac{103980997575196}{588898765297} a^{8} + \frac{386048833990633}{588898765297} a^{7} - \frac{734204497790813}{588898765297} a^{6} + \frac{904802596773420}{588898765297} a^{5} - \frac{816526140721754}{588898765297} a^{4} + \frac{510992666350410}{588898765297} a^{3} - \frac{223452171937029}{588898765297} a^{2} + \frac{59796572921437}{588898765297} a - \frac{3936223889923}{588898765297} \) (order $16$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23614.5375719 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{8})\), 4.0.4608.1 x2, 4.2.9216.1 x2, 4.0.18432.1 x2, 4.2.18432.3 x2, \(\Q(\zeta_{16})^+\), 4.0.2048.2, 8.0.339738624.8, 8.0.1358954496.9, \(\Q(\zeta_{16})\), 8.4.5435817984.1 x2, 8.0.5435817984.1 x2

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

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$