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Properties

Label 16.0.13604889600...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $18.13$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_4.D_4$ (as 16T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -184, 296, 404, -1668, 2378, -1896, 757, 48, -86, -98, 63, 10, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 10*x^13 + 63*x^12 - 98*x^11 - 86*x^10 + 48*x^9 + 757*x^8 - 1896*x^7 + 2378*x^6 - 1668*x^5 + 404*x^4 + 296*x^3 - 184*x^2 + 16)
 
gp: K = bnfinit(x^16 - 2*x^15 - 8*x^14 + 10*x^13 + 63*x^12 - 98*x^11 - 86*x^10 + 48*x^9 + 757*x^8 - 1896*x^7 + 2378*x^6 - 1668*x^5 + 404*x^4 + 296*x^3 - 184*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 8 x^{14} + 10 x^{13} + 63 x^{12} - 98 x^{11} - 86 x^{10} + 48 x^{9} + 757 x^{8} - 1896 x^{7} + 2378 x^{6} - 1668 x^{5} + 404 x^{4} + 296 x^{3} - 184 x^{2} + 16 \)

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:  \(136048896000000000000=2^{20}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.13$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{4} a^{8} + \frac{1}{3} a^{6} + \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{4} a^{9} + \frac{1}{3} a^{7} + \frac{1}{12} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{264} a^{14} - \frac{1}{132} a^{13} - \frac{1}{44} a^{12} + \frac{1}{44} a^{11} + \frac{7}{264} a^{10} - \frac{43}{132} a^{9} + \frac{2}{33} a^{8} + \frac{13}{66} a^{7} + \frac{43}{88} a^{6} + \frac{7}{33} a^{5} + \frac{7}{22} a^{4} + \frac{7}{66} a^{3} - \frac{1}{3} a + \frac{10}{33}$, $\frac{1}{2528420553627144} a^{15} + \frac{1478822526283}{1264210276813572} a^{14} - \frac{3937395561113}{114928206983052} a^{13} - \frac{28876355234803}{1264210276813572} a^{12} - \frac{14390182995283}{229856413966104} a^{11} + \frac{27703853102921}{421403425604524} a^{10} + \frac{1595424807629}{316052569203393} a^{9} + \frac{73866875854327}{316052569203393} a^{8} + \frac{1015045419242641}{2528420553627144} a^{7} - \frac{1049731126357}{210701712802262} a^{6} + \frac{244501857498713}{632105138406786} a^{5} + \frac{931449643933}{2899564855077} a^{4} - \frac{72533459078195}{316052569203393} a^{3} + \frac{11001523455493}{57464103491526} a^{2} - \frac{46473533853553}{105350856401131} a + \frac{18569914885655}{105350856401131}$

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:  \( -\frac{113228593}{1491719064} a^{15} + \frac{223667255}{1491719064} a^{14} + \frac{151524195}{248619844} a^{13} - \frac{540949465}{745859532} a^{12} - \frac{7154801365}{1491719064} a^{11} + \frac{3563010075}{497239688} a^{10} + \frac{1223706185}{186464883} a^{9} - \frac{1784062055}{745859532} a^{8} - \frac{28606393535}{497239688} a^{7} + \frac{209896644725}{1491719064} a^{6} - \frac{44764569493}{248619844} a^{5} + \frac{307129985}{2280916} a^{4} - \frac{8559260455}{186464883} a^{3} - \frac{120169955}{16951353} a^{2} + \frac{430292905}{62154961} a - \frac{46449853}{186464883} \) (order $6$)
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:  \( 5165.64894396 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4$ (as 16T30):

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 $C_4.D_4$
Character table for $C_4.D_4$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.2.2000.1, 4.2.18000.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.324000000.4, 8.0.7290000.1, 8.0.2916000000.5

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 sibling: 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
5Data not computed