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Properties

Label 16.0.11017296355...6624.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 13^{6}$
Root discriminant $23.86$
Ramified primes $2, 3, 13$
Class number $2$
Class group $[2]$
Galois group $C_2\times SD_{16}$ (as 16T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -16, 32, 240, 1808, -2120, 1216, -144, 844, -936, 520, -112, 68, -60, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 68*x^12 - 112*x^11 + 520*x^10 - 936*x^9 + 844*x^8 - 144*x^7 + 1216*x^6 - 2120*x^5 + 1808*x^4 + 240*x^3 + 32*x^2 - 16*x + 4)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 68*x^12 - 112*x^11 + 520*x^10 - 936*x^9 + 844*x^8 - 144*x^7 + 1216*x^6 - 2120*x^5 + 1808*x^4 + 240*x^3 + 32*x^2 - 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 60 x^{13} + 68 x^{12} - 112 x^{11} + 520 x^{10} - 936 x^{9} + 844 x^{8} - 144 x^{7} + 1216 x^{6} - 2120 x^{5} + 1808 x^{4} + 240 x^{3} + 32 x^{2} - 16 x + 4 \)

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:  \(11017296355467800346624=2^{32}\cdot 3^{12}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.86$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{12} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{12} a^{9} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{12} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{11} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{12} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{24} a^{12} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{12} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{24} a^{13} - \frac{1}{6} a^{6} + \frac{1}{4} a^{5} + \frac{1}{6} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{24} a^{14} - \frac{1}{6} a^{7} + \frac{1}{4} a^{6} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{10273215205808376} a^{15} + \frac{24663904195883}{1712202534301396} a^{14} - \frac{94358378682313}{10273215205808376} a^{13} + \frac{207547595926391}{10273215205808376} a^{12} - \frac{9742990896089}{856101267150698} a^{11} + \frac{23303724101444}{1284151900726047} a^{10} + \frac{8302822795801}{2568303801452094} a^{9} + \frac{3230429953171}{1712202534301396} a^{8} - \frac{420869486769105}{1712202534301396} a^{7} + \frac{305052224559715}{2568303801452094} a^{6} + \frac{604256810273039}{1712202534301396} a^{5} + \frac{2342819624512589}{5136607602904188} a^{4} - \frac{907720340830043}{2568303801452094} a^{3} - \frac{207753193827365}{856101267150698} a^{2} + \frac{113355608769506}{1284151900726047} a - \frac{210468807348967}{856101267150698}$

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:  \( \frac{98882757533909}{2568303801452094} a^{15} - \frac{1048203028225725}{3424405068602792} a^{14} + \frac{4165001616787177}{3424405068602792} a^{13} - \frac{23054729560743431}{10273215205808376} a^{12} + \frac{8493244119347285}{3424405068602792} a^{11} - \frac{1764345406346374}{428050633575349} a^{10} + \frac{33755491868846491}{1712202534301396} a^{9} - \frac{59809481147603833}{1712202534301396} a^{8} + \frac{25974756460644233}{856101267150698} a^{7} - \frac{15173599142137127}{5136607602904188} a^{6} + \frac{77387200905746985}{1712202534301396} a^{5} - \frac{134811522304609551}{1712202534301396} a^{4} + \frac{110729275129265381}{1712202534301396} a^{3} + \frac{13046212477508207}{856101267150698} a^{2} - \frac{543242919793654}{428050633575349} a - \frac{1421497360318315}{2568303801452094} \) (order $12$)
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:  \( 75200.3989622 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times SD_{16}$ (as 16T48):

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 $C_2\times SD_{16}$
Character table for $C_2\times SD_{16}$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.4.7488.1, 4.0.7488.1, \(\Q(\zeta_{12})\), 8.8.104963309568.1, 8.0.104963309568.1, 8.0.56070144.2

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$