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Properties

Label 16.0.16244794399...856.12
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 7^{8}$
Root discriminant $18.33$
Ramified primes $2, 3, 7$
Class number $4$
Class group $[4]$
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 32, 32, -512, 2108, -1680, 616, 244, -267, 120, 8, -48, 7, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^13 + 7*x^12 - 48*x^11 + 8*x^10 + 120*x^9 - 267*x^8 + 244*x^7 + 616*x^6 - 1680*x^5 + 2108*x^4 - 512*x^3 + 32*x^2 + 32*x + 16)
 
gp: K = bnfinit(x^16 - 4*x^13 + 7*x^12 - 48*x^11 + 8*x^10 + 120*x^9 - 267*x^8 + 244*x^7 + 616*x^6 - 1680*x^5 + 2108*x^4 - 512*x^3 + 32*x^2 + 32*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{13} + 7 x^{12} - 48 x^{11} + 8 x^{10} + 120 x^{9} - 267 x^{8} + 244 x^{7} + 616 x^{6} - 1680 x^{5} + 2108 x^{4} - 512 x^{3} + 32 x^{2} + 32 x + 16 \)

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:  \(162447943996702457856=2^{32}\cdot 3^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.33$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} - \frac{1}{8} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6712688} a^{14} - \frac{95235}{1678172} a^{13} - \frac{175011}{3356344} a^{12} - \frac{364821}{1678172} a^{11} + \frac{48647}{6712688} a^{10} - \frac{653273}{1678172} a^{9} - \frac{1514845}{3356344} a^{8} - \frac{96973}{839086} a^{7} - \frac{425643}{6712688} a^{6} - \frac{12520}{419543} a^{5} + \frac{1259929}{3356344} a^{4} + \frac{391505}{839086} a^{3} - \frac{430585}{1678172} a^{2} + \frac{289011}{839086} a - \frac{8196}{419543}$, $\frac{1}{5804090498002192} a^{15} + \frac{55989719}{5804090498002192} a^{14} - \frac{2580758385735}{42677136014722} a^{13} - \frac{1825091963785}{32977786920467} a^{12} - \frac{986934716423253}{5804090498002192} a^{11} - \frac{816647103205771}{5804090498002192} a^{10} + \frac{120391114665621}{725511312250274} a^{9} - \frac{1026214968935}{5735267290516} a^{8} - \frac{2046043122655667}{5804090498002192} a^{7} + \frac{1116693031757939}{5804090498002192} a^{6} - \frac{271640700115331}{1451022624500548} a^{5} + \frac{256121056737097}{1451022624500548} a^{4} + \frac{261526455571731}{1451022624500548} a^{3} - \frac{54983111037979}{725511312250274} a^{2} - \frac{5521133421981}{65955573840934} a + \frac{286408470895815}{725511312250274}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{657902765375}{341417088117776} a^{15} + \frac{29483157225}{7422110611256} a^{14} + \frac{46643403033}{85354272029444} a^{13} + \frac{5380549897}{674737328296} a^{12} - \frac{10002187632489}{341417088117776} a^{11} + \frac{20227429123779}{170708544058888} a^{10} - \frac{17465403588953}{85354272029444} a^{9} - \frac{3506627825985}{15518958550808} a^{8} + \frac{334345724247837}{341417088117776} a^{7} - \frac{250364397695293}{170708544058888} a^{6} - \frac{33094229260911}{85354272029444} a^{5} + \frac{978182720128611}{170708544058888} a^{4} - \frac{876972509521189}{85354272029444} a^{3} + \frac{735098662445223}{85354272029444} a^{2} - \frac{2043766997484}{1939869818851} a + \frac{27288892490197}{42677136014722} \) (order $8$)
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:  \( 4819.71504316 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_4$ (as 16T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{21}) \), 4.2.4032.1, 4.2.1792.1, 4.2.16128.1, 4.2.448.1, \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(i, \sqrt{42})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{2}, \sqrt{21})\), 8.0.12745506816.8, 8.0.12745506816.16, 8.0.12745506816.19, 8.0.260112384.5, 8.0.3211264.1, 8.4.796594176.2, 8.4.12745506816.5

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$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}$
$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}$