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Properties

Label 16.0.42286532693...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{4}\cdot 7^{4}$
Root discriminant $16.85$
Ramified primes $2, 3, 5, 7$
Class number $2$
Class group $[2]$
Galois group $C_2\times Q_8:C_2^2$ (as 16T69)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 0, 0, 112, -69, 44, 128, -140, 57, 116, -120, 20, 71, -68, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 68*x^13 + 71*x^12 + 20*x^11 - 120*x^10 + 116*x^9 + 57*x^8 - 140*x^7 + 128*x^6 + 44*x^5 - 69*x^4 + 112*x^3 + 49)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 68*x^13 + 71*x^12 + 20*x^11 - 120*x^10 + 116*x^9 + 57*x^8 - 140*x^7 + 128*x^6 + 44*x^5 - 69*x^4 + 112*x^3 + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 68 x^{13} + 71 x^{12} + 20 x^{11} - 120 x^{10} + 116 x^{9} + 57 x^{8} - 140 x^{7} + 128 x^{6} + 44 x^{5} - 69 x^{4} + 112 x^{3} + 49 \)

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:  \(42286532693852160000=2^{32}\cdot 3^{8}\cdot 5^{4}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.85$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{63} a^{13} - \frac{1}{21} a^{12} - \frac{4}{63} a^{11} + \frac{10}{63} a^{10} - \frac{26}{63} a^{9} - \frac{19}{63} a^{8} + \frac{1}{7} a^{7} + \frac{1}{9} a^{6} - \frac{2}{21} a^{5} - \frac{2}{63} a^{4} + \frac{3}{7} a^{3} + \frac{11}{63} a^{2} - \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{945} a^{14} - \frac{4}{189} a^{12} - \frac{31}{315} a^{11} + \frac{67}{945} a^{10} + \frac{232}{945} a^{9} - \frac{398}{945} a^{8} + \frac{121}{315} a^{7} - \frac{118}{945} a^{6} + \frac{52}{189} a^{5} + \frac{37}{135} a^{4} - \frac{349}{945} a^{3} - \frac{1}{105} a^{2} + \frac{14}{135} a - \frac{14}{135}$, $\frac{1}{777391965} a^{15} - \frac{2794}{13176135} a^{14} + \frac{151373}{22211199} a^{13} + \frac{23554522}{777391965} a^{12} - \frac{1358371}{22211199} a^{11} + \frac{3234808}{22211199} a^{10} - \frac{43337516}{155478393} a^{9} - \frac{292097189}{777391965} a^{8} - \frac{74981716}{777391965} a^{7} - \frac{42145217}{777391965} a^{6} - \frac{29156689}{86376885} a^{5} + \frac{121324169}{259130655} a^{4} + \frac{14782924}{155478393} a^{3} - \frac{309861613}{777391965} a^{2} - \frac{4273811}{37018665} a + \frac{34995709}{111055995}$

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:  \( \frac{495748}{22211199} a^{15} - \frac{2564924}{13176135} a^{14} + \frac{136796332}{155478393} a^{13} - \frac{364376144}{155478393} a^{12} + \frac{3081833828}{777391965} a^{11} - \frac{2804634017}{777391965} a^{10} + \frac{410169013}{777391965} a^{9} + \frac{2820901673}{777391965} a^{8} - \frac{2387067518}{777391965} a^{7} - \frac{1197820387}{777391965} a^{6} + \frac{103392568}{17275377} a^{5} - \frac{1217672513}{259130655} a^{4} + \frac{949916164}{777391965} a^{3} + \frac{1847806129}{777391965} a^{2} - \frac{103734703}{37018665} a + \frac{187545464}{111055995} \) (order $24$)
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:  \( 10473.1096153 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times Q_8:C_2^2$ (as 16T69):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 34 conjugacy class representatives for $C_2\times Q_8:C_2^2$
Character table for $C_2\times Q_8:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{24})\), 8.0.6502809600.6, 8.0.406425600.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{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}$