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Properties

Label 16.0.634562281237118976.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 7^{8}$
Root discriminant $12.96$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -16, 48, -72, 90, -60, 76, -82, 51, -66, 48, -30, 29, -20, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 12*x^14 - 20*x^13 + 29*x^12 - 30*x^11 + 48*x^10 - 66*x^9 + 51*x^8 - 82*x^7 + 76*x^6 - 60*x^5 + 90*x^4 - 72*x^3 + 48*x^2 - 16*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 + 12*x^14 - 20*x^13 + 29*x^12 - 30*x^11 + 48*x^10 - 66*x^9 + 51*x^8 - 82*x^7 + 76*x^6 - 60*x^5 + 90*x^4 - 72*x^3 + 48*x^2 - 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 12 x^{14} - 20 x^{13} + 29 x^{12} - 30 x^{11} + 48 x^{10} - 66 x^{9} + 51 x^{8} - 82 x^{7} + 76 x^{6} - 60 x^{5} + 90 x^{4} - 72 x^{3} + 48 x^{2} - 16 x + 4 \)

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:  \(634562281237118976=2^{24}\cdot 3^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.96$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{18} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{6} a^{8} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{5}{18} a^{4} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{9} - \frac{1}{9} a^{7} - \frac{1}{6} a^{5} - \frac{1}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{10} - \frac{1}{9} a^{8} - \frac{1}{6} a^{6} - \frac{1}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9} a$, $\frac{1}{4498866} a^{15} - \frac{6309}{249937} a^{14} + \frac{62078}{2249433} a^{13} + \frac{6517}{749811} a^{12} + \frac{35855}{4498866} a^{11} + \frac{175742}{2249433} a^{10} + \frac{59564}{2249433} a^{9} + \frac{36247}{249937} a^{8} - \frac{638159}{4498866} a^{7} - \frac{86881}{2249433} a^{6} + \frac{769309}{2249433} a^{5} + \frac{37969}{2249433} a^{4} + \frac{76489}{749811} a^{3} - \frac{881624}{2249433} a^{2} + \frac{193622}{2249433} a + \frac{14520}{249937}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{437611}{749811} a^{15} + \frac{3263029}{1499622} a^{14} - \frac{4725248}{749811} a^{13} + \frac{43068155}{4498866} a^{12} - \frac{29734760}{2249433} a^{11} + \frac{55523383}{4498866} a^{10} - \frac{16859464}{749811} a^{9} + \frac{15236117}{499874} a^{8} - \frac{13205903}{749811} a^{7} + \frac{171184049}{4498866} a^{6} - \frac{71372129}{2249433} a^{5} + \frac{86145181}{4498866} a^{4} - \frac{95545142}{2249433} a^{3} + \frac{62385139}{2249433} a^{2} - \frac{29340598}{2249433} a + \frac{1659106}{749811} \) (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:  \( 1538.34886394 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

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 $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(i, \sqrt{21})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{7})\), 4.0.1008.2 x2, 4.0.1008.1 x2, 4.2.9408.1 x2, 4.2.9408.2 x2, 8.0.49787136.1, 8.0.16257024.2 x2, 8.0.88510464.1 x2, 8.0.796594176.7, 8.0.796594176.14, 8.0.49787136.3 x2, 8.4.796594176.1 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$