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Properties

Label 16.0.11819246862...7024.9
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 $C_2^2:D_4$ (as 16T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 16, 96, 112, 240, 608, 384, -420, -576, -8, 240, 88, -24, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 - 24*x^13 + 88*x^12 + 240*x^11 - 8*x^10 - 576*x^9 - 420*x^8 + 384*x^7 + 608*x^6 + 240*x^5 + 112*x^4 + 96*x^3 + 16*x^2 + 4)
 
gp: K = bnfinit(x^16 - 16*x^14 - 24*x^13 + 88*x^12 + 240*x^11 - 8*x^10 - 576*x^9 - 420*x^8 + 384*x^7 + 608*x^6 + 240*x^5 + 112*x^4 + 96*x^3 + 16*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} - 24 x^{13} + 88 x^{12} + 240 x^{11} - 8 x^{10} - 576 x^{9} - 420 x^{8} + 384 x^{7} + 608 x^{6} + 240 x^{5} + 112 x^{4} + 96 x^{3} + 16 x^{2} + 4 \)

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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{5}$, $\frac{1}{40} a^{14} - \frac{1}{40} a^{13} + \frac{1}{40} a^{12} - \frac{1}{40} a^{11} - \frac{1}{40} a^{9} - \frac{1}{20} a^{8} - \frac{1}{2} a^{7} - \frac{1}{20} a^{6} - \frac{7}{20} a^{5} - \frac{1}{4} a^{4} - \frac{7}{20} a^{3} + \frac{2}{5} a^{2} - \frac{7}{20} a - \frac{1}{10}$, $\frac{1}{1623218000} a^{15} + \frac{279932}{101451125} a^{14} - \frac{44925761}{811609000} a^{13} + \frac{4634682}{101451125} a^{12} - \frac{2769209}{811609000} a^{11} - \frac{4335311}{101451125} a^{10} + \frac{4086663}{162321800} a^{9} - \frac{10027377}{202902250} a^{8} + \frac{362078619}{811609000} a^{7} + \frac{3125368}{20290225} a^{6} - \frac{67013153}{405804500} a^{5} - \frac{5296369}{101451125} a^{4} - \frac{138213959}{405804500} a^{3} + \frac{28086354}{101451125} a^{2} - \frac{72546479}{405804500} a - \frac{2245712}{101451125}$

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:  \( -\frac{964}{7825} a^{15} + \frac{7011}{62600} a^{14} + \frac{29821}{15650} a^{13} + \frac{18909}{15650} a^{12} - \frac{195841}{15650} a^{11} - \frac{588069}{31300} a^{10} + \frac{134821}{6260} a^{9} + \frac{1826541}{31300} a^{8} - \frac{38807}{7825} a^{7} - \frac{383439}{6260} a^{6} - \frac{192797}{7825} a^{5} + \frac{65631}{7825} a^{4} - \frac{72811}{7825} a^{3} - \frac{8547}{15650} a^{2} + \frac{59043}{15650} a - \frac{13779}{15650} \) (order $4$)
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:  \( 8127.25924519 \)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), 4.0.512.1, 4.0.4608.1, 4.2.18432.2, 4.2.18432.1, \(\Q(i, \sqrt{6})\), 8.0.1358954496.8, 8.0.339738624.7, 8.0.1358954496.5

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$